Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann

793

Jean Renault

A Groupoid Approach to C*-Algebras

Springer-Verlag Berlin Heidelberg New York 1980

Author

Jean Renault Departement de Mathematiques Faculte des Sciences 45 Orleans - La Source France

AMS Subject Classifications (1980): 22 D 25, 46 L 05, 54 H 15, 54 H 20

ISBN 3-540-09977-8 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-0997?-8 Springer-Verlag NewYork Heidelberg Berlin

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin Heidelberg 1980 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210

CONTENTS

Introduction

I .

2.

3.

4.

Chapter I : LOCALLY COMPACT GROUPOIDS

Def in i t ions and Notation

Locally Compact Groupoids and Haar Systems

Quasi- lnvariant Measures

Continuous Cocycles and Skew-Products

1.

2.

3.

4.

5.

Chapter I I : THE C*-ALGEBRA OF A GROUPOID

The Convolution Algebras Cc(G,~ ) and C*(G,o)

Induced Representations

Amenable Groupoids

The C*-Algebra of an r-Discrete Principal Groupoid

Automorphism Groups, KMS States and Crossed Products

Chapter I I I : SOME EXAMPLES

1. Approximately-Finite Groupoids

2. The Groupoids 0 n

Page

I

5

5

16

22

35

47

48

74

86

97

109

121

121

138

Appendix : The Dimension Group of the GICAR Algebra

References

Notation Index

Subject Index

148

151

155

157

INTRODUCTION

The interp lay between ergodic theory and von Neumann algebra theory goes back to

the examples of non-type I factors which Murray and von Neumann obtained by the group

measure construction [54]. A natural and probably de f i n i t i ve point of view which

joins both theories has recently been exposed by P. Hahn [45]. I t uses the notion of

measure groupoid, introduced by G. Mackey "to bring to l i gh t and exp lo i t certain

apparently far reaching analogies between group theory and ergodic theory"

( [53], p.187). In par t icu lar , the group measure algebra may be regarded as the von

Neumann algebra of the regular representation of some pr incipal measure groupoid.

Moreover, most of the properties of the algebra may be interpreted in terms of the

groupoid. The same standpoint is adopted by J. Feldman and C.Moore [31], in the

framework of ergodic equivalence re la t ions. Besides, they characterize abstract ly

the von Neumann algebras ar is ing from the i r construction.

I t is natural to expect that topological l oca l l y compact groupoids play a simi-

la r role in the theory of C*-algebras. The notions of topological and of Lie

groupoid were introduced by Ehresmann for applications to d i f f e ren t i a l topology and

geometry. More recent in terest in topological groupoids has come from the theory of

fo l i a t ions ([10] ,p.273). I t seems to be the d i f f e ren t i a l geometry point of view,

rather than Mackey's v i r tua l group point of view which aroused J. Westman's in terest

in groupoids and led him to the construction of convolution algebras of groupoids,

f i r s t in the t rans i t i ve (and loca l l y t r i v i a l ) case [75] and then in the non-transi t ive

pr incipal case [77]. However the relevance to the theory of induced representations

is also apparent in [7~ . Convolution algebras of transformation groups had already

been used for some time [16,37]. The main works about transformation group C*-algebras,

by Effros and Hahn [24] and by Zel ler -Meier [80] , appeared at about the same time

as Westman's a r t i c l e . Although the i r main purpose is to construct in teres t ing exam-

ples of C*-a lgebras, Effros and Hahn also give some resul ts on the structure of a

transformation group C*-a lgebra. This goal is more apparent in Ze l ler -Meier 's work,

which is more d i r ec t l y motivated by group representation theory. Most of the l a t t e r

work about transformation group C*-algebras concerns i t s e l f wi th the structure

theory of these algebras ( fo r example [39]).

The s tar t ing point of th is work is a theorem of S. S t r~ t i l # and D.Voiculescu

about approx imate ly- f in i te (or AF) C*-algebras [ 6~ . Generalizing the method of

L.Garding and A. Wightman [3 4 fo r studying factor representations of the canonical

anticommutation re la t ions of mathematical physics, they show that every AF C*-a lgebra

can be diagonalized and use a d iagonal izat ion to study i t s s t ructure and i t s repre-

sentations. In our se t t ing , th is amounts to saying that every AF C*-a lgebra is the

C*-a lgebra of a pr inc ipal groupoid (3.1.15).

The construct ion (2.1) of the C*-a lgebra of a groupoid is modelled a f te r the

construct ion of the C*-a lgebra of a transformation group given by Effros and Hahn.

Since a l oca l l y compact groupoid does not necessari ly have a Haar system, (Westman

uses the term of l e f t invar iant continuous system of measures), needed to define

the convolut ion product, and since such a Haar system need not be unique, (although

some resul ts about existence and uniqueness of Haar systems can be found in K. Seda's

a r t i c l es [67,68], we consider l o ca l l y compact groupoids with a f ixed Haar system.

The case of r -d iscre te groupoids, which generalize discrete transformation groups,

deserves special a t ten t ion , because i t includes a l l our examples. An r -d iscre te

groupoid has a Haar system i f and only i f i t s range map is a local homeomorphism, and,

i f th is is the case, i t is a scalar mul t ip le of the counting measures system (1 .2 .8 . ) .

In the general case, but under sui table hypotheses, we show that the strong Morita

equivalence class of the C *-algebra does not depend on the choice of the Haar system

(2.2.11).

The theory of group C*-algebras suggests many general izat ions. In pa r t i cu la r ,

one expects a correspondence between un i tary representations of the groupoid and non-

degenerate representations of i t s C*-a lgebra. This is established (2.1.23) under a

rather technical condit ion which w i l l often be needed, namely the existence of s u f f i -

c i en t l y many non-singular Borel G-sets ( de f i n i t i on 1.3.27). I t is also possible to

induce a representation from a closed subgroupoid (2.2.9). We give a de f i n i t i on of

amenabil i ty in section 3 of chapter 2. I t develops that the C*-a lgebra of an amena-

ble groupoid concides with the reduced C*-a lgebra, obtained by considering only the

representations induced from the un i t space (2.3.2). Moreover, using some of

R. Zimmer's ideas about amenable measure groupoids [82 ,8~ , i t is eas i ly shown that

th is C*-a lgebra is nuclear (2.3.5) .

From our point of view, the most in teres t ing groupoids are pr inc ipal groupoids.

Their C*-algebras appear as genuine general izat ions of matr ix algebras. We have

looked for a character izat ion of these algebras s imi lar to the condit ion given by

Feldman and Moore for algebras over an ergodic equivalence re la t ion . The notion of

Cartan subalgebra we give (2.4.13) is rather r e s t r i c t i v e and not as congenial as the

corresponding notion for von Neumann algebras. In pa r t i cu la r , we show by an example

(3.1.17) that a regular maximal se l f -ad jo in t abelian subalgebra which is the image

of a unique condi t ional expectation need not be a Cartan subalgebra. The correspon-

dence between closed two-sided ideals of the reduced C*-a lgebra of a pr inc ipa l

groupoid and the closed invar iant subsets of i t s un i t space is established in the

r -d iscre te case (2.4.6) .

A continuous homomorphism (also cal led a one-cocycle) from a l oca l l y compact

groupoid to a l oca l l y compact abelian group defines a continuous homomorphism of the

dual group into the automorphism group of the C*-a lgebra of the groupoid (2.5.1).

Moreover many one-parameter automorphism groups of the AF C *-algebras considered in

mathematical physics (e.g. gauge automorphism group, dynamical groups) arise in th is

fashion (examples 3.1.6 and 3.1.10). The groupoid point of view is pa r t i cu l a r l y well

suited to t he i r study. For example, the Connes spectrum of such an automorphism group

is the asymptotic range of the cocycle (2.5.8) and the crossed product C*-a lgebra is

the C*-a lgebra of the skew-product (2.5.7). Besides, the KMS condit ion for states

may be replaced by a condit ion much closer to the or ig ina l Gibbs Ansatz character iz ing

equi l ibr ium states (2.5.4) . We use groupoids to derive par t i cu la r but important cases

of some theorems of D. 01esen and G.K. Pedersen [5~ about s imp l i c i t y and p r i m i t i v i t y

of crossed product C ~-a lgebras as well as the main resul ts of O.Brat te l i [ ~ . Another

appl icat ion of groupoid C ~-algebras is the study of the C * -a lgebra of the b i cyc l i c

semi-group and of the Cuntz C ~-algebras (3.2).

A number of fundamental problems have not been touched in th is work. As we have

seen e a r l i e r , groupoids have been introduced for two reasons. One is the " v i r t ua l

group" point of v iew,we have not even given a de f i n i t i on of s i m i l a r i t y for l oca l l y

compact groupoids with Haar system. The other is the appl icat ion to d i f f e r e n t i a l

geometry, in par t i cu la r to the theory of f o l i a t i ons ; we have not made any mention

of the work of A. Connes in th is d i rec t ion . These topics must await fu r ther develop-

ment in the future.

The author wishes to express indebtness to Marc Rief fe l for numerous and f r u i t -

fu l suggestions and to Paul Muhly for a careful reading of the manuscript. He would

l i ke to thank P. Hahn, who taught him about groupoid algebras and J.Westman for some

unpublished material he gave him.

CHAPTER 1

LOCALLY COMPACT GROUPOIDS

The f i r s t chapter sets up the framework of th is study. To gain some motivation

for the de f in i t ions which are given there, the reader can look simultaneously at the

examples of the th i rd chapter. I t is also useful to keep in mind the example of

transformation groups, which is recal led below, and which suggests most of the

terminology.

The f i r s t section gives the algebraic sett ing of the theory. The two main

concepts are groupoids and inverse semi-groups. The de f i n i t i on of a l oca l l y compact

groupoid with Haar system is introduced in the second section. The th i rd section

deals with the notion of quasi - invar iant measure, and a general izat ion of i t , the

KMS condit ion, The results thereof w i l l be of great use in the second chapter. Some

elementary properties of one-cocycles are studied in the fourth section. Given a one-

cocycle, one can build the skew-product groupoid, and a basic question is to deter-

mine i t s structure in terms of the cocycle. An essential tool is the asymptotic range

of the cocycle, which is the topological analog of Krieger's asymptotic ra t io set in

ergodic theory (see [ 3 ~ , I , de f i n i t i on 8.2).

1. Def in i t ions and Notation

We shall use the de f i n i t i on of a groupoid given by P. Hahn in [4 4 (de f in i t i on

1.1). I t is essent ia l ly the same as the one used by J.Westman in [7~ and the one

used by A. Ramsay in [61].

1.1. Def in i t ion : A groupoid is a set G endowed with a product map (x,y) ~ xy :

G 2 ÷ G where G 2 is a subset of G x G cal led the set of composable pairs, and an

-1 i n v e r s e map x ~ x

( i)

( i i )

( i i i )

(iv)

: G ~ G such t h a t the f o l l o w i n g r e l a t i o n s are s a t i s f i e d :

( x - l ) - I = x

, G 2 ( x , y ) ( y , z ) ~ => ( x y , z ) , ( x , y z ) c G 2 and ( x y ) z = x ( y z )

( x - l , x ) E G 2 and i f ( x , y ) c G 2, then x -1 (xy ) = y

( x , x - I ) c G 2 and i f ( z , x ) E G 2, then ( z x ) x -1 = z

I f x ~ G, d ( x ) = x - l x i s the domain o f x and r ( x ) = xx -1 i s i t s range. The p a i r

( x , y ) i s composable i f f the range o f y i s the domain o f x . G O = d(G) = r (G) i s the

u n i t space o f G, i t s e lements are u n i t s in the sense t h a t x d ( x ) = x and r ( x ) x = x.

Un i t s w i l l u s u a l l y be denoted by u, v , w w h i l e a r b i t r a r y e lements w i l l be denoted

by x , y , z.

I f A and B are subsets o f G, one may form the f o l l o w i n g subsets o f G :

A - I {x ~ G : x - I = E G }

AB = {z E G : x ~ A, y ~ B : z = xy } .

A g roupo id G is sa id to be p r i n c i p a l i f the map ( r , d ) f rom G i n t o G O x G O is

o n e - t o - o n e , i t i s sa id to be t r a n s i t i v e i f the map ( r , d ) i s o n t o .

, = d - l ( v ) G u = GUn G and For u, v , ~ GO GU = r - 1 ( u ) ' Gv ' v v

G(u) = G u u ' which i s a g roup , i s c a l l e d the i s o t r o p y group a t u.

The r e l a t i o n u ~ v i f f G u # @ is an e q u i v a l e n c e r e l a t i o n on the u n i t soace G O . V

I t s e q u i v a l e n c e c lasses are c a l l e d o r b i t s and the o r b i t o f u i s denoted [ u ] . GO/G

denotes the o r b i t space. A g r o u p o i d i s t r a n s i t i v e i f f i t has a s i n g l e o r b i t .

1 .2 . Examples :

a. T r a n s f o r m a t i o n g roups

Suppose t h a t the group S ac t s on the space U on the r i g h t . The image o f the

p o i n t u by the t r a n s f o r m a t i o n s i s denoted u -s . We l e t G be U x S and d e f i n e the

f o l l o w i n g g roupo id s t r u c t u r e : ( u , s ) and ( v , t ) are composable i f f v = u . s ,

( u , s ) ( u . s , t ) = ( u , s t ) , and (u , s ) -1 = ( u . s , s - 1 ) . Then r ( u , s ) = ( u ,e ) and d ( u , s ) =

( u ' s , e ) . The map ( u , e ) ~ u i d e n t i f i e s G O w i t h U. The t e r m i n o l o g y o f o r b i t s comes

f rom t h i s example. Then G i s p r i n c i p a l i f f S ac ts f r e e l y , and t r a n s i t i v e i f f S ac t s

t r a n s i t i v e l y .

b. The groupoid G 2

The set G 2 of composable elements may be given the f o l l o w i n g groupoid s t r uc tu re :

(x ,y ) and ( y ' , z ) are composable i f f y~ = xy, ( x , y ) (xy ,z ) = ( x , y z ) , and (x ,y ) -1 =

( xy , y -Z ) .

Then r 2 ( x ,y ) = ( x , r ( y ) ) = ( x , d ( x ) ) and d 2 ( x , y ) = ( x y , d ( x y ) ) . The map

x ~ ( x , d ( x ) ) i d e n t i f i e s the u n i t space of G 2 w i th G. The groupoid G 2 is p r i n c i p a l .

One may no t ice tha t i t comes from the ac t ion o f G on i t s e l f . I t is t r a n s i t i v e i f f

G is a group.

c. Equivalence r e l a t i o n s

Let R be the graph of an equivalence r e l a t i o n on a set U. We give to R the

f o l l o w i n g groupoid s t ruc tu re : (u ,v) and ( v ' ,w ) are composable i f f v' = v, (u ,v)

(v,w) = (u ,w) , and (u ,v) - I = ( v ,u ) . Then, r ( u , v ) = (u,u) and d(u ,v ) = ( v , v ) . The u n i t

space of R is the diagonal and may be i d e n t i f i e d w i th U. R is a p r i n c i p a l groupoid.

Conversely, i f G is a p r i n c i p a l groupoid, ( r , d ) i d e n t i f i e s G wi th the graph of the

equivalence r e l a t i o n ~.

d. Group bundle

A group bundle G is a groupoid such tha t fo any x ~ G, d(x) = r ( x ) . A group

bundle is the union of i t s i so t ropy groups G(u). Here, two elements may be composed

i f f they l i e in the same f i b e r . Given any groupoid G, G' = {x ~ G : d(x) = r ( x ) }

is a group bundle. We ca l l i t the i so t ropy group bundle of G. I t is reduced to the

un i t space of G i f f G is p r i n c i p a l .

1.3. D e f i n i t i o n : Let G and H be groupoids. A map ~ : G ÷ H, is a homomorphism i f

f o r any (x ,y ) ~ G 2, ( ~ ( x ) , ¢ (y ) ) ~ H 2 and ¢(x) ¢(y) = ¢(xy) . Then ¢(u) ~ H 0

i f u c G O . ¢0 : G O ÷ H 0 denotes the r e s t r i c t i o n o f ¢ to the u n i t spaces.

¢2 : G 2 ÷ H 2 is the ma9 ¢2(x ,y ) = ( ¢ ( x ) , ¢ ( y ) ) ; i t is a homomorphism. Two homomor-

phisms ¢,¢ : G ÷ H are s i m i l a r ( w r i t e ¢ ~ ~ ) i f there ex i s t s a func t ion e : G O ÷ H

such tha t ( e~ r ) ( x ) ¢(x) = ~(x) (eod) (x ) f o r any x ~ G. Groupoids G and H are ca l l ed

s i m i l a r ( w r i t e G ~ H) i f there e x i s t homomorphisms ¢ : G ÷ H and ~ : H ÷ G such tha t

¢ o ~ and ~ o ¢ are s i m i l a r to i d e n t i t y isomorphisms.

Before g i v ing a r e s u l t of Ramsay [61] (theorem 1.7, p. 260) which i l l u s t r a t e s

t h i s no t ion , we need a d e f i n i t i o n .

1.4, D e f i n i t i o n : Let G be a groupoid, E a subset of G O ; G I = {x ~ G : r ( x ) e E ~E

and d(x)E E} is a subgroupoid o f G w i th u n i t space E ; GIE is ca l l ed the reduct ion

of G by E.

1 .5 .P ropos i t i on : Let G be a groupoid, E a subset o f G O which meets each o r b i t in

G O ," then GIE ~ G.

1.6. D e f i n i t i o n : Let G be a groupoid, A a group and c : G ÷ A a homomorphism, the

skew-product G(c) is the groupoid G x A where : ( x ,a ) and (y ,b) are composable

i f f x and y are composable and b = ac (x ) , ( x , a ) ( y , a c ( x ) ) = ( x y , a ) , and (x ,a ) -1 =

( x - l , a c ( x ) ; r ( x , a ) = ( r ( x ) , a ) , d ( x , a ) = ( d ( x ) , a c ( x ) ) . I t s u n i t space is G O x A.

A basic example of skew-product is the f o l l o w i n g . Let s be a t rans fo rmat ion of

the space U in to i t s e l f and l e t f be a func t ion on U wi th values in anabel ian group A.

On the space U x A, de f ine the t rans fo rmat ion t by ( u , a ) t = (us,a + 1~(u)). Let us

def ine the groupoid G o f s as the groupoid associated w i th the corresponding t rans-

format ion group ( U , Z ) and def ine s i m i l a r l y the groupoid of t . We leave to the reader

to check tha t the groupoid of t is the skew-product of the groupoid G of s by the

homomorphism c : G -~ A obta ined from f by the ru les

n-1 c (u ,n) = Z f ( u t i ) f o r n > 1,

O

c(u,O) = O, and

c (u , -n ) = - c (u ,n ) f o r -n < -1.

Another impor tant way of bu i l d i ng up new groupoids from o ld ones is the semi-

d i r e c t product .

1.7. D e f i n i t i o n : Let G be a groupoid , l e t A be a group and l e t ~ : A + Aut(G) be a

homomorphism. We w r i t e x-a = ~ ( a - l ) ] (x) f o r a ~ A and x ~ G. The s e m i - d i r e c t

product G x A is the groupoid G x A where (x ,a ) and (z ,b ) are composable i f f z = y-a

wi th x and y composable, ( x , a ) ( y . a ,b) = ( xy ,ab ) , and (x ,a) -1 = (x -1 • a, a - l ) .

Then, r ( x , a ) = ( r ( x ) , e ) and d (x ,a ) = (d(x) • a , e ) . The u n i t space may be i d e n t i f i e d

w i th G O .

An example of s e m i - d i r e c t product is the groupoid associated w i th a t ransforma-

t i o n group (U,A). In t h i s case G = U is reduced to i t s un i t space. When G is a group,

1.7 is the usual no t ion of s e m i - d i r e c t product .

There is a natura l ac t ion o f A on the skew-product G(c) , namely the homomorphism

def ined by the formula m(a) (x ,b ) = (x ,ab) and there is a natura l homomorphism c

o f the s e m i - d i r e c t product G x A i n to A, def ined by the formula c (x ,a ) = a.

1.8. P ropos i t i on : With above n o t a t i o n ,

( i ) G(c) x A is s i m i l a r to G and

( i i ) (G x A)(c) is s i m i l a r to G.

Proof : One may apply 1.5. For example, to prove ( i ) , one observes tha t the subset

E = G O x {e} o f the u n i t space G O x A of G(c) x A meets each o r b i t . For f u r t h e r re- c~

ference, l e t us w r i t e down e x p l i c i t l y the s i m i l a r i t y homomorphisms :

( i ) Define ~ from G(c) x A to G by ~ (x ,a ,b ) = x, de f ine ~ from G to G(c) x A C~ C~

by ~(x) = ( x , e , c ( x ) ) and def ine 0 from G O x A to G(c) x A by e (u ,a ) = (u ,e ,a -1) and

check tha t ~ o ~ (x) = x and e [ r ( x , a , b ) ] ( x , a , b ) = ~o~p ( x ,a ,b ) e [ d ( x , a , b ) ] .

( i i ) Define ~ from (G x A) (c ) to G by ~ ( x ,a ,b ) = x - b -1 def ine ~ from G to

(G x A)(c) by ¢(x) = ( x ,e ,e ) and def ine e from G O x A to (G x A)(c) by 0(u ,a) =

(u • a - l , a , e ) and check t ha t ~ o ~(x) = x and 0 [ r ( x , a , b ) ] ( x , a , b ) = ~ o ¢ ( x , a , b )

e [ d ( x , a , b ) ] . Q.E.D.

Together w i th the not ion of groupo id , the not ion of inverse semi-group plays an

impor tant ro le in t h i s work. The d e f i n i t i o n given below, as wel l as some elementary

p r o p e r t i e s , can be found in [11 ] , page 28, or [1 ] .

10

1.9. D e f i n i t i o n : An inverse semi-group is a s e t ~ endowed w i th an a s s o c i a t i v e b ina -

ry o p e r a t i o n , noted m u l t i p l i c a t i v e l y , and an inve rse map s ÷ s -1 : ~ ÷ ~ such t h a t the

f o l l o w i n g r e l a t i o n s are s a t i s f i e d : s s - l s = s and s - l ss -1 = s - I .

Then the i nve rse map is an i n v o l u t i o n . I f s ~ , d(s) = s - l s i s the domain o f

s and r ( s ) = ss -1 is i t s range. The set o f idempotent elements is denoted by g O Two

idempotent elements commute. The r e l a t i o n e ~ f i f f e f = e is an o rder r e l a t i o n on gO

which makes i t i n t o an i n f s e m i - l a t t i c e .

The r e l a t i o n between groupoids and inverse semi-groups is g iven by i n t r o d u c i n g

the no t i on o f G-set o f a g roupo id .

1 . 1 0 . D e f i n i t i o n : Le t G be a groupo id . A subset s o f G w i l l be ca l l ed a G-set i f

the r e s t r i c t i o n s o f r and d to i t are one- to -one . E q u i v a l e n t l y , s is a G-set i f f ss - I

and s - I s are conta ined in G O .

Let g be the se t o f G-sets o f G. We note t h a t s , t ~ g => s t e g and

-1 s e g => s ~ g . These opera t i ons make g i n t o an inverse semi-group. Note t ha t the

no ta t i ons d(s) and r ( s ) agree w i th the prev ious ones.

A G-set s de f ines va r ious maps as f o l l o w s :

( i ) f o r x on G w i th d(x) ~ r ( s ) , the element xs o f G is de f ined by {xs } = {x }s

( t h i s makes sense) ;

( i i ) f o r x in G w i th r ( x ) ~ d ( s ) , the element sx of G is de f ined by {sx } = s {x } ;

( i i i ) f o r u in r ( s ) , the element u -s in d(s) is de f ined by u . s = d (us) . These

n o t a t i o n s w i l l be used s y s t e m a t i c a l l y . The map u ~ u • s : r ( s ) ÷ d(s) w i l l be c a l l e d

the G-map assoc ia ted w i t h the G-set s. The reader should not have any t r o u b l e to check

t ha t

× ( s t ) = ( x s ) t ; ( t s ) x = t ( s x ) ; (xs) -1 = s-Zx - I ;

u • ( s t ) = (u . s) • t

where, w i th our conven t i on , x ( s t ) is de f ined by { x ( s t ) } = { x } s t f o r the G-sets s and

t and s i m i l a r l y ( t s ) x is de f ined by { ( t s ) x } = t s { x } .

11

To help understanding what G-sets mean, l e t us look at the case of a t r a n s f o r -

mation group (U,S). Any element s of the group S def ines the f o l l o w i n g G-set o f the

associated groupoid G : s = { ( u , s ) : u e U}. I t s domain and i t s range are U. The

associated G-map is the t rans fo rmat ion u ~ u • s and there is no ambigui ty in the

no ta t i ons . The map from S to the set of G-sets above def ined is an inverse semi-group

homomorphism. I t is one- to-one but usua l l y not onto. Note tha t in the case o f a group,

tha t i s , when U is reduced to one po in t , the G-sets are exac t l y the elements of S.

J. Westman has developed in [7~ a cohomology theory f o r groupoids which extends

the usual group cohomology theory ; i t is reproduced here.

Suppose tha t C is some category. A map p from a set A onto a set A 0 such t ha t

each f i b e r p - l ( u ) is an ob jec t of C w i l l be ca l led a C-bundle map and A w i l l be

ca l l ed C-bundle . For example, a group bundle in the sense of 1.2.d is a C -bund le

where C is the category of groups and any such C-bund le is a group bundle. Let A be

a C-bund le w i th bundle map p : A ÷ A O. Wri te A u = p - l ( u ) . Iso(A) = {isomorphisms

#u,v : Av ÷ Au : u ,v , E A O} has a natura l s t r uc tu re of groupoid : #u,v and ~v ' ,w

i f f v' = v - then t h e i r product is ~u,v °#v,w' and 4 -1 is the i so - are composable ' U~V

morphism inverse of ~u,v" The b i j e c t i o n idu , u ~ u i d e n t i f i e s the un i t space of Iso(A)

and A O. Iso(A) is ca l l ed the isomorphism groupoid o f the C-bund le A.

1.11. D e f i n i t i o n : Let G be a groupoid. A G-bundle (A,L) is a C -bund le A toge ther

w i th a homomorphism L : G ÷ I s o ( A ) such tha t L 0 : G O ÷ A 0 is a b i j e c t i o n . (We w i l l

o f ten i d e n t i f y G O and AO). When C i s the category of abe l ian groups, one speaks o f a

G-module bundle.

Given a G-module bundle (A ,L ) , one can form the f o l l o w i n g cochain complex. Let

us f i r s t de f ine G n fo r any n ~ N. The sets G 0, G I = G and G 2 have a l ready been d e f i -

ned. For n ~ 2, G n is the set of n-uples (x 0 . . . . . Xn_l) c Gx.. .xG such tha t f o r

i = 1 . . . . . n - l , x i is composable w i th i t s l e f t neighbor. A n-cochain is a func t ion f

from G n to A which s a t i s f i e s the cond i t ions

( i ) po f (x 0 . . . . . Xn_l) = r(XO) and

( i i ) i f n > 0 and fo r some i = O , . . . , n - 1 , x i c G O , then f ( x 0 . . . . . x i . . . . . Xn_l)

12

A 0 E . The set Cn(G,A) of n-cochains is an abel ian group under pointwise add i t ion . The

an> n+l sequence 0 ÷ cO(G,A) ÷ CI(G,A) . . . . . Cn(G,A) - - C (G,A) . . . . . where ~Of(x) =

n L(x) f~d(x) - f o r ( x ) and an( f (x 0 . . . . . Xn) = L (xo ) f ( x I . . . . . Xn) + Z (-1) i

i= l f ( x 0 . . . . . x i _ i x i . . . . . Xn_l) + (-1) n+l f ( x 0 . . . . . Xn_l) fo r n > O, is a cochain complex.

1.12. D e f i n i t i o n : The group of n-cocycles of th is complex w i l l be denoted by Zn(G,A),

the group o f n-coboundaries w i l l be denoted by Bn(G,A)and the n- th cohomology group

Zn(G,A)/Bn(G,A) w i l l be denoted by Hn(G,A).

A sect ion for a G-bundle (A,L) is a func t ion f from A 0 to A such that pof(u)

= u, where p is the bundle map. A sect ion f is said to be i nva r i an t i f L(x) fod(x)=

f o r ( x ) fo r every x ~ G. The set of sections w i l l be denoted by F(A) and the set of

i nva r i an t sect ions by I~G(A). I f (A,L) is a G-module bundle, cO(G,A) = F(A) and

HO(G,A) = rG(A ).

A one-cocycle c ~ ZI(G,A) is a one-cochain f from G to A which s a t i s f i e s f (xy)=

L ( x ) f ( y ) + f ( x ) . In p a r t i c u l a r , i f A is a constant bundle, that i s , each f i b e r A u is

equal to a f i xed abe l ian group B, and i f G acts t r i v i a l l y on A, tha t i s , L(x) is the

i d e n t i t y map of B fo r every x, a one--cocycle f c ZI(G,A) is a homomorphism of G in to

B. In the case of a constant bundle A as above wi th t r i v i a l ac t ion , we wr i t e ZI(G,B)

instead of ZI(G,A).

We may also consider one-cocycles with values in a not necessar i ly abel ian

group. In th is case, (A,L) is a G-bundle where A is a group bundle. We def ine ZI(G,A)

= { f : G ÷ A : f (xy ) = f ( x ) [ L ( x ) f ( y ) ] } , BI(G,A) = { f : G ÷ A : there ex is ts b : G O ÷ B

such that f ( x ) = [b o r ( x ) ] - l [ L ( x ) b o d ( x ) ] } and the equivalence r e l a t i o n on ZI(G,A) :

f ~, g i f f there ex is ts b : G O ÷ B such that f ( x ) = [ b o r ( x ) ] - 1 (x) [ L ( x ) b o d ( x ) ] .

As fo r groups, two-cocycles are re la ted to groupoid extensions :

1.13. D e f i n i t i o n : Let (A,L) be a G-module bundle, noted m u l t i p l i c a t i v e l y . An

extension of A by G is an exact sequence of groupoids

A 0 ÷ A-~i>E~J>G ÷ G O (we also w r i t e ( E , i , j ) )

compatible wi th the act ion of G on A, in the sense that there ex is ts a sect ion k fo r j

such that

( i ) k(u) = u (A O, E 0 and G O

( i i ) k(x) i ( a ) k(x) -1 = i ( L ( x ) a

13

are i d e n t i f i e d )

fo r any (a ,x ) e A x G w i th p(a) = d (x ) .

Two extensions ( E , i , j ) and ( E ' , i , j ' ) are equ iva len t i f there ex is ts an isomor-

phism # : E ÷ E' such tha t i ' = #oi and j = j ' o # . The set of equ iva len t classes of

extensions w i th the Baer sum is an abe l ian group denoted Ext(A,G).

1.14. Propos i t i on : H2(G,A) = Ext(A,G).

Sketch of the proof : Given ~ ~ Z2(G,A), l e t E = { ( a , x ) c A x G : p(a) = r ( x ) } . o

I t s groupoid s t ruc tu re is given by

(a ,x ) and (b ,y ) are composable i f f x and y are ; then

( a , x ) ( b , y ) = ( a ( m ( x ) b ) ~ ( x , y ) , x y )

and (a ,x ) - I = ( ( m ( x - l ) a - 1 ) ~ ( x - I , x ) - l , x - I ) .

Define i ( a ) = ( a ,p (a ) ) and j ( a , x ) = x and note tha t k(x) = ( r ( x ) , x ) is a covar ian t

sec t ion . I t is r e a d i l y v e r i f i e d tha t ( E , i , j ) is an extension and tha t i t s class

depends only on the class of o.

Conversely, i f ( E , i , j ) is an extension of A by G and k is a covar ian t sec t ion ,

then ~ def ined by i ( ~ ( x , y ) ) =k (x )k ( y ) k ( xy ) -1 is a 2-cocycle in Z2(G,A). I t s c lass is

not a f fec ted by another choice of sect ion or an equ iva len t ex tens ion. F i n a l l y @:

(a ,x ) ~ i ( a ) k ( x ) : E ~ E sets up an equivalence of E and E.

The t r i v i a l extension is the s e m i - d i r e c t product of A and G.

Let us f i n a l l y note tha t two s i m i l a r groupoids have same cohomology groups w i th

c o e f f i c i e n t s in a t r i v i a l constant module bundle. E x p l i c i t l y , l e t # : G ÷ H and ~ :

H ÷ G be two h a l f - s i m i l a r i t i e s ; ~o~ ~ id G and ~o ~ ~ id H. The maps f ÷ fo~n :

Cn(G,A) ~ Cn(H,A) and g ~ go#n : Cn(H,A) -~ Cn(G,A) g ive isomorphisms of the cohomo-

logy groups.

A cohomology theory f o r inverse semi-groups may be given along the same l i n e s .

Suppose t h a t C is some category. Let A 0 be a se t . The set 2 AO of a l l subsets of A O,

when ordered by i nc l us i on , is a category : there is an arrow V ÷ U p rec i se l y when

V c U. A C - s h e a f A based on A 0 is a con t rava r i an t func to r U ÷A U on 2 AO t o C ( the

14

morphism "~U ~J{V corresponding to V c U should thought o f as the r e s t r i c t i o n morphism).

A pa r t i a l isomorphism ~ of J{ is a b i j e c t i o n # : V + U, where V and U are subsets

of A 0 together wi th isomorphisms # :~{V' ÷ ~ ( V ' ) ' f o r any V ' c V, compatible wi th

the r e s t r i c t i o n morphisms, tha t i s , such that f o r V " c V ' , the f o l l ow ing diagram com-

mutes i V ' * ~ ( V ' ) .

~ V l ' ' ~t (~) ( Vii )

Two pa r t i a l isomorphisms # and #' may be composed : we have # : V -~ U and #' : V' ÷ U' ;

we l e t V" be #,-1 (U'n V) and U" be #(U'n V) ; #" = #o#' is the b i j e c t i o n V" ÷ U"

obtained by composing # and #' ; and fo r V c V" we def ine #" :J{V_ ÷ ~ # " ( V ) by com-

' ~ < The inverse of a pa r t i a l isomorphism is def ined in pos ing~v ~ '> j {# , (V ) # o# , (V).

the obvious fash ion. These operat ions make jso(Y~) = { p a r t i a l isomorphisms of~4} i n to

an inverse semi-group, that we ca l l the isomorphism inverse semi-group of the C-sheafs { .

1.15. D e f i n i t i o n : Let ~ be an inverse semi-group. A g -shea f (~,£) is a C-sheaf

together w i th a homomorphism £: g ÷ Jso(~) such tha t [0 : gO ÷ 2Ao is an i n j e c t i o n .

We l e t gn be g x . . . x g n times fo r n > 1 and gO be as before. Given a g -sheaf

(~{, £) of abel ian groups, one can form the fo l l ow ing cochain complex. A n-cochain is

a func t ion f from g n to~{ which s a t i s f i e s the condi t ions

( i ) f(So,S I . . . . . Sn_1) e ~{ r(SoS 1 .--Sn_ 1) ;

( i i ) f is compatible wi th the r e s t r i c t i o n maps, tha t i s , i f U = r (s 0 s I . . .Sn_ l )

and V = r ( t 0 t I . . . t n_ l ) where t i = eis fo r some idempotent element e i then

f ( t o ' t l . . . . . tn-1) ~ ~V is the r e s t r i c t i o n of f ( s ,s . . . . . . s l ) c~ . , to V ; and A o ± n-~ u ' 0

( i l l ) f o r n • O, f (s 0 . . . . . s i . . . . . Sn_l) ~ 2 whenever s i is an idempotent element.

The set c n ( g , J { ) of n-cochains is an abel ian group under pointwise add i t i on . The

sequence

~n cn+l ( 0 + c O ( g , ~ ) ~ c Z ( g , ~ ) . . . . . + c n ( g , . ~ ) ....... > ~ , ~ ) ÷

where 6Of(s) = £ (s ) f od(s) - fo r (s )

and ~nf(s 0 . . . . . Sn) = £ (So) f (s I . . . . . Sn) n

+ i ! i ( - 1 ) i f ( s o . . . . . s i -1 si . . . . . Sn)

+ (-1) n+l f (s 0 . . . . Sn_l ) ,

is a cochain complex.

15

1.16, Def in i t ion : The group of n-cocycles and the group of n-coboundaries of this

complex w i l l be denoted respect ively by z n ( ~ , ~ ) and by B n ( ~ , ~ ) . The n-th cohomo-

logy group z n ( g , ~ ) / B n ( g , ~ ) w i l l be denoted H n ( g , ~ ) .

Before giving the next de f in i t i on , l e t us remark that ~ = u ~ U, where U runs

overdO AO = 2 , has a structure of inverse semi-group, where for a C~U and b c~ V,

a + b is the element o f ~ UnV obtained by adding up the res t r i c t ions of a and b to UnV.

1.17. Def in i t ion : Let ( ~ , £) be a g-sheaf of abelian groups, noted m u l t i p l i c a t i -

vely. An extension o f ~ by ~ is an exact sequence of inverse semi-groups

y~O ÷j~ i> 8 j> g ÷ ~0 (we also wr i te ( ~ , i , j ) )

compatible with the action of ~ on~ in the sense that there exists a section k for j

such that

( i )

( i i )

( i i i )

k(e) = e for e ~ gO (~cO 80 and gO are iden t i f ied)

k(s) i (a) k(s) - I = i ( £ ( s ) a ) for (a,s) ~ J~x~.

k(es) = ek(s) and k(se) = k(s)e for e c ~ 0 s ~ .

Two extensions ( 8 , i , j ) and ( 8 ' , i ' , j ' ) are equivalent i f there exists an isomorphism

: 8 + 8' such that i ' = #oi and j = j'o@. The set of equivalence classes of exten-

sions with the Baer sum is an abelian group denoted Ext ( ~ , ~ ) a n d jus t as before,

Ext ( ~ , ~ ) is isomorphic to H2(j~, ~) .

1.18. F ina l l y , we note the re lat ionship between the cohomology of a groupoid G and

the cohomology of the inverse semi-group of i t s G-sets ,~ . Let (A,L) be a G-module

bundle. One forms the fol lowing ~-sheaf of abelian groups based on A O, ( ~ , £ ) . For

UcAO,J~U = {sections of A defined on U} with i t s addi t ive structure ; for VcU, the

morphism ~ ÷ ~ is the usual res t r i c t i on map. The homomorphism £ : ~÷ Jso(d~) is

defined by : £(s) is the b i ject ion d(s)÷ r(s) which sends u into u . s - I and for

V c d ( s ) and U = Vs - I C r ( s ) , £(s) : ~ ' V ÷ ~ is given by £(s) h(u) = L(us) h(u . s)

for h E~ V. A cochain f e Cn(G,A) defines a cochain f c c n ( ~ , j ~ ) . Namely

f(So,S 1 . . . . . Sn_l) is the section of A defined on r(s 0 s I . . . Sn_l) by

f(So,S I . . . . . Sn_l) (u) = f(USo,(U • So)S I . . . . . (u - SoS 1 . . . Sn_2)Sn_l). I t is compa-

t i b l e with the res t r i c t i on maps. The map f ~ f commutes with the coboundary opera-

tors, 8n~ = (~nf)-.Therefore ' i f f E Zn(G,A) (rasp Bn(G,A)), then f ~ Z n (~,d~)

16

(resp B n ( ~ , ~ ) ) . Conversely, given g E c n ( ~ , ~ ) , we may def ine f ~ Cn(G,A) by

f(Xo,X I . . . . . Xn_l) = g( {x } , {x I } . . . . . {x n 1 } ) (r(XO)) where {Xo} , { x 1} . . . . . ~x n 1 ) are 0 - ~ -

considered as G-sets. Then g = f . In conclusion c n ( ~ , ~ ) ~ Cn(G,A) ; zn(~,~) ~ zn(m,A) ;

B n ( ~ , ~ ) ~ Bn(G,A) ; H n ( ~ , ~ ) ~ Hn(G,A). We w i l l use a t opo log i ca l vers ion of t h i s

r e s u l t in 2.14.

2. Loca l l y Compact Groupoids and Haar Systems.

The d e f i n i t i o n of a t opo log i ca l groupoid and i t s immediate consequences can be

found in [79] , [26] page 23 and [68] page 26.

2.1. D e f i n i t i o n : A topo log i ca l groupoid cons is ts of a groupoid G and a topology

compat ib le w i th the groupoid s t ruc tu re :

- i ( i ) x ~ x : G ~ G is continuous

( i i ) ( x ,y ) ~> xy : G 2 -- G is continuous where G 2 has the induced topology from

G x G.

- 1 , Consequences : x ~ x is a homeomorphism ; r and d are continuous ; i f G is

Hausdorf f , G O is closed in G ; i f G O is Hausdorf f , G 2 is c losed in G x G. G O is both

a subspace of G and a quo t i en t of G (by the map r) ; the induced topology and the quo-

t i e n t topology co inc ide .

We w i l l only consider t opo log i ca l groupoids whose topology is Hausdorff and,

w i th the except ion of sect ion 4, l o c a l l y compact. We w i l l usua l l y use Bourbaki 's

theory of i n t e g r a t i o n on l o c a l l y compact spaces [ 5 , 6 , 7 ] .

I f X is a l o c a l l y compact space, Cc(X ) denotes the l o c a l l y convex space of

complex-valued continuous func t ions w i th compact suppor t , endowed w i th the induc t i ve

l i m i t topo logy .

2.2. D e f i n i t i o n : Let G be a l o c a l l y compact groupoid. A l e f t Haar system fo r G

cons is ts of measures {~u, u e G O } on G such tha t

( i ) the support supp ~u of the measure ~u is G u,

17

( i i ) ( con t i nu i t y ) fo r any f ~ Cc(G),u ~ ~ ( f ) (u ) = f fd~ u is cont inuous, and

( i i i ) ( l e f t invar iance) fo r any x ~ G and any f ~ Cc(G ), f f ( x y ) d~d(X)(y) =

f f (Y)dAr(X) (y ) .

This is Westman's d e f i n i t i o n ([77] p.2) of a l e f t i n va r i an t continuous system

of measures. I t d i f f e r s from Seda's d e f i n i t i o n ([68] p .27 ) . i n two respects : no

measure on the un i t space is given and con t i nu i t y is required ; th is l as t assumption

is a ra ther severe r e s t r i c t i o n on the topology of G. In Section 4 of [68] and

theorem 2 of [67], Seda gives condi t ions under which con t i nu i t y holds au tomat i ca l l y ;

i t seems pre ferab le here to assume i t as par t of the d e f i n i t i o n .

The fo l l ow ing resu l ts are easy consequences of the d e f i n i t i o n (c f . [77 ] 1.3 , ! . 4 ) .

2.3. Propos i t ion : ~: Cc(C ) -~ Cc(GO ) is a continuous su jec t ion .

2 .4 . Propos i t ion : Let G be a l o c a l l y compact groupoid with a l e f t Haar system.

Then r : G -~ G O is an open map, and the associated equivalence r e l a t i o n on the un i t

space is open.

2.5. Examples : !

(a) A l o c a l l y compact t ransformat ion group G = U x S has a d is t ingu ished l e f t

Haar system : ~u = ~u x ~, where ~u is the point-mass at u and ~ a l e f t Haar measure

for S.

(b) I f G is a l o c a l l y compact groupoid, then G 2 wi th the topology induced from

G x G is also a l o c a l l y compact groupoid. I f { u} is a l e f t Haar system fo r G, then

{ (~2)x} is a l e f t Haar system fo r G 2 where

f f d(~2) x = f f ( x , z ) d~ d(x) (z) fo r f ~ Cc(G2 ).

For example, i f G is a group, G 2 = G x G. As a groupoid, i t is the groupoid associated

with the t ransformat ion group (G,G) where G acts on i t s e l f by t r ans l a t i on . I t s l e f t

Haar system is 6 x x ~, where ~ is a l e f t Haar measure fo r G, as in example a.

(c) Let G be a l o c a l l y compact p r inc ipa l groupoid. The map d : G u ~ [u] is a

b i j e c t i o n which gives to [u] a l o c a l l y compact topology, which can be d i f f e r e n t from

the topology induced from G O . An a l t e rna te d e f i n i t i o n fo r a l e f t Haar system on G is :

18

a system of measures {m[u] ' u c G O } where

( i ) m[u] is a measure on [u] of support [u]

( i i ) f o r any f ~ Cc(G),u ~ f f (u ,v )dm[u ] (V ) is continuous (G is viewed as a sub-

set of G O x GO).

These d e f i n i t i o n s are equ iva lent : i f {Xu } is g iven, ~[u] = d ~ U depends only on [u]

and s a t i s f i e s ( i ' ) and ( i i ' ) ; conversely i f {~ [u ] } is g iven, {~u} is a l e f t Haar

system, vlhere f fd~ u = f f (u ,v )dm[u ] ( v ) .

(d) Let G be a l o c a l l y compact group bundle, that i s , a l o c a l l y compact groupoid

which is a group bundle in the sense of 1.2.d. Then a l e f t Haar system, i f i t ex i s t s ,

is e s s e n t i a l l y unique in the sense tha t two l e f t Haar systems {~u} and {v u} d i f f e r by

a continuous pos i t i ve funct ion h on G O : xu = h(u) u. The iso t ropy group bundle G' =

{x e G : d(x) = r ( x ) } of a l o c a l l y compact groupoid G is closed, hence l o c a l l y

compact. In the case where G is a t ransformat ion group, the existence of a l e f t Haar

system on G' is the assumption made in [3~ (see beginning of the f i r s t sect ion page

886) to determine the topo log ica l s t ruc ture of the space of a l l i r r educ ib l e induced

representat ions of G.

(e) Let G be a l o c a l l y compact group. The set S of subgroups of G becomes a

compact Hausdorff space when equipped wi th F e l l ' s topology [32]. G = {(K,x) : K E S,

x c K} c S x G wi th the topology induced from S x G and the groupoid s t ruc ture :

(K,x) and (L,y) are composable i f f K = L, (K,x) (K,y) = (K,xy) , (K,x) -1 = (K,x -1) is a

l o c a l l y compact group bundle, that we may ca l l the subgroups bundle of G. I t is shown

in ~2] that a l e f t Haar system (~K) ex i s t s . I t is e s s e n t i a l l y unique by d. For each

K c S, ~K is a l e f t Haar measure fo r K.

2.6. D e f i n i t i o n : A l o c a l l y compact groupoid is r - d i sc re te i f i t s un i t space is an

open subset.

2.7. Lemma : Let G be an r -d i sc re te groupoid.

( i ) For any u e G O , G u and G u are d isc re te spaces.

( i i ) I f a Haar system ex i s t s , i t is e s s e n t i a l l y the counting measures system.

( i i i ) I f a Haar system ex i s t s , r and d are local homeomorphisms.

19

Proof :

( i )

in G v, {x}

( i i )

An x in G v def ines a homeomorphism y ~ xy : G v ÷ G u - since #v} is ooen U ~ • " '

is open in G u.

Let {~u} be a l e f t Haar system. Since G u is d i sc re te and ~u has support G u,

every po in t in G u has p o s i t i v e ~U-measure. Let g = ~,(×GO ) , where XG 0 is the characte-

r i s t i c func t ion of G O . I t is continuous and p o s i t i v e . Replacing ~u by g(u)-1~ u, we

may assume tha t ~U({u}) = 1 f o r any u. Then by invar iance , ~V({x} ) = 1 f o r any

G v X ~ . u

( i i i ) We assume, as we may, tha t xu is the counting measure on G u. Let x be a

po in t of G. A compact neighborhood V o f x meets G u in f i n i t e l y many po in ts x i

i = 1 . . . . . n. I f x i ~ x, there ex i s t s a compact neighborhood V' of x contained in V,

which does not conta in x . . Therefore, we may assume tha t GunV = { x } . Then ~ r ( x ) ( v )= 1. l

By c o n t i n u i t y o f the Haar system, we may assume tha t ~U(v) = 1 f o r any u ~ r (V) . This

shows tha t r : V ÷ G O is i n j e c t i v e , hence a homeomorphisms onto r (V) .

2.8. Propos i t ion : For a l o c a l l y compact groupoid G, the f o l l ow ing p rope r t i es are

equ iva len t :

( i ) G is r - d i s c r e t e and admits a l e f t Haar system,

( i i ) r : G + G O is a loca l homeomorphism,

( i i i ) the product map G 2 ÷ G is a loca l homeomorphism, and

( i v ) G has a base of open G-sets.

Proof :

( i ) ~ > ( i i ) This has been shown in 7 ( i i i ) .

( i i ) ~ > ( i i i ) I f ( x ,y ) E G 2, we may choose a compact neighborhood U of x and

a compact neighborhood V of y such tha t r l v and d iv are homeomorphisms onto t h e i r I

images ; U x V n G 2 is then a compact neighborhood of (x ,y ) on which the product map

is i n j e c t i v e .

x ' y ' = x"y" = > r ( x ' ) = r ( x " ) ~ > x ' = x"

and d ( y ' ) = d (y " ) = > y ' = y " .

( i i i ) ~ ( i v ) I f x E G and U is a neighborhood of x, we may f i nd open sets V

and W such tha t x e V c U, x -1 U -1 W c and the r e s t r i c t i o n of the product map to

20

V x W is i n jec t i ve . SoV n W -1 is the des i red open G-set.

( i v ) =-=> ( i i ) C lear .

( i v ) -~->(i) The groupoid G is r - d i s c r e t e : f o r any u c G O , there is an open

G-set s such tha t u e r ( s ) = s s - l c G O and by ( i i i ) ss -1 is open in G.

Let ~u be the count ing measure on G u and f be in C c (G). Using a p a r t i t i o n of the

i d e n t i t y , one can w r i t e f as a f i n i t e sum of func t ions supported on open G-sets s,

Therefore i t is enough to consider a func t ion f whose support is contained in an open

G-set s. Then ~ ( f ) ( u ) = ~u( f ) = f (us) : ~ ( f ) is cont inuous.

Q.E.D.

2.9, Co ro l l a r y : A l o c a l l y compact groupoid G is r - d i s c r e t e and admits a l e f t Haar

system i f f G 2 is r 2 - d i s c r e t e and admits a l e f t Haar system.

2.10. D e f i n i t i o n : Let G be an r - d i s c r e t e groupoid. I t s ample semi-group ~ i s the

semi-group of i t s compact open G-sets.

This te rmino logy , in t roduced by W. Kr ieger in [5 4 , w i l l be j u s t i f i e d at the

end of the sect ion. The case of i n t e r e s t is when G admits a cover of compact open

G-sets. Then G has a base of open G-sets, w i th sub-base {Us : U open subset o f G O

and s c ~ } , t he re fo re G admits a l e f t Haar system. We do not know i f there e x i s t r -

d i s c r e t e groupoids which have a Haar system but do not have a cover of compact open

G-sets. I f G has a cover of compact open G-sets, i t is complete ly descr ibed by (G O , ~ )

in the sense tha t i t s groupoid s t ruc tu re as wel l as i t s topology may be recovered from

G O , ~ and the map r . I f x ~ s, w i th s e ~ , x -1 is def ined by s -1 { r ( x ) } = x - I . I f

x c s, y c t w i th s, t E ~ and d(x) = r ( y ) , xy is def ined by {xy} = { r ( x ) } s t . We have

j u s t seen tha t {Us : U open subset of G O , s E ~ } is a sub-base f o r the topology o f G.

Let us descr ibe next the r - d i s c r e t e p r i n c i p a l groupoids which admit a cover of

compact open G-sets.

2.11. D e f i n i t i o n : Let U be a l o c a l l y compact space and s a p a r t i a l homeomorphism

o f U, def ined on a compact open subset r ( s ) onto a compact open subset d (s ) . Let us

say tha t s is r e l a t i v e l y f ree i f i t s set of f i xed points {uc r ( s ) : u • s = u} is

(compact and) open. Let us say tha t an inverse semi-group ~ o f p a r t i a l homeomorphisms

21

def ined on compact open subsets of U acts r e l a t i v e l y f r e e l y i f each s E~ is re la -

t i v e l y f ree.

2.12. D e f i n i t i o n : Let U be a l o c a l l y compact space and ~an inverse semi-group o f

p a r t i a l homeomorphisms def ined on compact open subsets o f U. Let us say tha t ~ is

ample i f

( i ) f o r any compact open set e in U, the i d e n t i t y map id e belongs to ~ .

( i i ) f o r any f i n i t e fami ly ( s i ) i=1 . . . . . n in ~ such that r ( s i ) n r ( s j ) =

and d ( s i ) n d ( s j ) = ~ fo r i # j , there ex is ts s in ~denoted by ~s i such tha t u • s =

u • s i f o r u ~ r ( s i ) .

2.13. Proposi t ion : Let U be a l o c a l l y compact space and ~ an inverse semi-group

o f p a r t i a l homeomorphisms def ined on compact open subsets of U. Let G be the

p r inc ipa l groupoid associated wi th the equivalence r e l a t i o n

u ~ v i f f there ex is ts s ~ ~ : u = v • s

Then the fo l l ow ing proper t ies are equ iva len t .

( i ) G has a s t ruc ture o f r -d i sc re te groupoid with a cover o f compact open

G-sets such that U becomes i t s un i t space and i t s ample semi-group is the ample in -

verse semi-group generated by ~ .

( i i ) ~ acts r e l a t i v e l y f r e e l y on U.

Proof :

( i ) ~ > ( i i ) Let s and t be two compact open G-sets of G. Then s n t is a com-

pact open G-set of G. Thus, i f s ~

{u ~ r (s ) : u • s = u} = s n r (s ) s compact open in G O = V.

( i i ) : > ( i ) For s ~ ~ , l e t s = { (u,us) : u c r ( s ) } . We def ine on G the topo lo -

gy which has as sub-base {Vs : V open in U and s ~ 3 } . I t makes G in to a r -d i sc re te

groupoid admit t ing a cover o f compact-open sets ,namely~.The induced topology on GO=u

is i den t i ca l to the o r i g i n a l one. F i n a l l y , l e t s be a compact open G-set. I t may be

covered by f i n i t e l y many open G-sets in ~ . Hence there ex is ts a f i n i t e fami ly (s i )

i = 1 . . . . . n i n ~ and a f i n i t e fami ly (Ui) i = 1 . . . . . n o f compact open sets o f U such n

that U i n Uj = ~, U i • s i n Uj • sj = ~ fo r i # j and s = U Uis i . i = l

Q.E.D.

$2

2.14. In the topo log ica l se t t i ng , we make the fo l l ow ing adjustments to the cohomolo-

gy theory given in the f i r s t sect ion (see [79] p.24).

(a) In 1.11, we requi re tha t A be a l o c a l l y compact group bundle and we re-

qu i re that fo r any continuous sect ion u ~ a u of p : A-~ A O, the funct ion x ~ L(X)ad(x)

should be continuous.

(b) We give to G n the topology induced from the product topology on Gx...xG

n-times and consider continuous cochains only. I t w i l l be i m p l i c i t that Zn(G,A),

Bn(G,A) and Hn(G,A) re fe r to the continuous cohomology.

I f G is an r -d i sc re te groupoid which admits a cover o f compact open G-sets,

the resu l ts o f 1.18 are s t i l l va l i d when ~ is in te rp re ted as the ample semi-group

of G. Given g ~ c n ( ~ , ~ ) (notat ions of 1.18), we def ine f e Cn(G,A) by

f(Xo,X 1 . . . . . Xn_l) = g(So,S 1 . . . . . Sn_ l ) ( r (xo) ) where So,S 1 . . . . . Sn_ I ~ a n d x 0 ~ s O ,

x I ~ s I . . . . . Xn_ 1 ~ Sn_ 1. By assumption, there ex i s t s O , s I . . . . . Sn_ 1 wi th these

proper t ies . Moreover, the cond i t ion that g be compatible wi th the r e s t r i c t i o n maps

shows that f is wel l def ined. F i n a l l y f is continuous since i t s r e s t r i c t i o n to

s O x SlX...XSn_ 1 is continuous. Thus Hn(G,A) ~ H n ( ~ , ~ ) .

3. Quas i - lnvar ian t Measures

Let G be a l o c a l l y compact groupoid wi th l e f t Haar system {~u}. Let ~u = (~u)- I

be the image of ~u by the inverse map x ÷ x -1. Then {~u } is a r i g h t Haar system.

3.1. D e f i n i t i o n : Let u be a measure on G O . The measure on G induced by ~ is

= f~Ud~(u). The measure on G 2 induced by ~ is 2 = f~ux~U d~(u). The image of

by the inverse map is - 1 = f~ud~(u)"

These measures are wel l def ined since the system {~u} o f measures on G and

{~u×~U} 2 the system o f measures on G 2 are ~-adequate (Bourbaki [6] 3.1) ; v is a lso

the measure on G 2 induced by - 1 wi th respect to the Haar system 2.5.b.

23

3.2. D e f i n i t i o n : A measure ~ on G O is said to be muas i - invar ian t i f i t s induced

measure ~ is equ iva len t to i t s inverse - I . A measure belonging to the class of

is also quas i - i nva r i an t ; we say tha t the class is i nva r i an t .

I f G is second countable and ~ is a quas i - i nva r i an t measure on G O , then (G,C),

where C is the class o f ~, is a measure groupoid in the sense of P.Hahn [44] p.15

and (v,u) is a Haar measure fo r (G,C) ( d e f i n i t i o n 3.11 p. 39). Most of the resu l ts

and techniques o f th i s sect ion can be found in [44] and in [61] .

The cohomology theory fo r measure groupoids is developed in [76] ; the

d isc re te p r inc ipa l case is studied thoroughly in [31]. The re levant fac t here is tha t

to each quas i - i nva r i an t measure is associated a 1-cocycle wi th values in fR~, whose

class depends on the measure class only .

3 .3 .Propos i t ion : Let ~ be a quas i - i nva r i an t measure on G O and D a l o c a l l y ~ - i n te -

grable pos i t i ve funct ion such that v = Dv - I , then

( i ) f o r 2 a.e. (x ,y ) ~D(xy) = D(x)D(y) and

fo r ~ a.e. x~ D(x - I ) = D(x) - I ;

( i i ) i f u' = g~ where g is a l o c a l l y ~ - in teg rab le pos i t i ve func t ion , D' =

(g o r)D (g o d) - I s a t i s f i e s v' = D'v ' -1

Proof :

( i )

D 2 (x ,y ) = D(xy)D(x) - I are versions of the Radon-Nikodym d e r i v a t i v e ~

gives the f i r s t asser t ion.

( i i ) S t ra ight forward.

(see also [44], theorem 3.1, p. 31) One shows that D2(x,y) = D(y) and

d~2 ; th i s

(dv2) -1

Q.E.D.

This p ropos i t ion shows that the Radon-Nikodym de r i va t i ve of ~ wi th respect to

- i (def ined a .e . ) is a one-cocycle With values i n lR : in the sense o f [76] §3 V

and that i t s class depends on the class of ~ only .

3.4. D e f i n i t i o n : Let ~ be a quas i - i nva r i an t measure on G O ," (a vers ion o f ) the

Radon-Nikodym d e r i v a t i v e D = dv is ca l led the modular func t ion (or the Radon- -1 d~

24

Nikodym d e r i v a t i v e ) of u.

I f G is a group, the po in t mass a t e i s , up to a sca la r m u l t i p l e , the only

q u a s i - i n v a r i a n t measure on G O = {e } . I t s modular func t ion in the sense o f 3.4 equals

a .e . the modular f unc t i on of the group.

I t w i l l be convenient f o r l a t e r purpose to choose ~ p a r t i c u l a r symmetric measure

in the class of u , where symmetric means equal to i t s inverse (the inverse of a

measure on G is i t s image under the inverse map). We choose ~O = D-1/2 ~ and ca l l i t

the symmetric measure induced by u.

3.5. D e f i n i t i o n : Let ~ be a q u a s i - i n v a r i a n t measure on G O . A measurable set A in

G O is almost i n v a r i a n t (w i th respect to u) i f f o r v a.e. x , r ( x ) c A i f f d(x) ~ A.

The measure ~ is ca l led ergodic i f every almost i n v a r i a n t measurable set is nul l or

conu l l .

Let X and Y be l o c a l l y compact spaces and p a continuous map from X onto Y.

I f X is q -compact , i t is poss ib le to def ine the image p . C of a measure class C on X :

one chooses a p r o b a b i l i t y measure ~ in the class of C and def ines p . C as the class

o f p . ~, where p . ~ ( E ) = ~ ( p - l ( E ) ) ; p , C depends only on the class of C. As i t is

eas ie r to deal w i th measures ra the r than w i th measure c lasses, one i~roduces the not ion

o f pseudo-image of a measure (see [6 ] ) : a pseudo-image of an a r b i t r a r y measure u on

X is a measure in the image p . C o f the class C of p .

3.6. P ropos i t i on : Let pbe a measure on G O and [u]

induced measure v . Then

( i ) [~] is a q u a s i - i n v a r i a n t ; and

( i i ) ~ is q u a s i - i n v a r i a n t i f f ~ ~ [~ ] .

Proof :

( i )

[ v ] ( f ) = O i f f f o r [~] a.e. v, ~v( f )= 0 ;

i f f f o r v a.e. x and ~d(x) a .e .

i f f f o r p a.e. u, ~u a.e. x and

be a pseudo-image by d o f the

Let Iv] = /~v d [ p ] ( v ) and f be a non-negat ive measurable func t ion . Then

y , f ( y ) = 0 ;

~d(x) a.e. u, f ( y ) = 0

25

( i i )

3 .7. D e f i n i t i o n :

i f f f o r ~ a .e . u, ~u a .e . x and x-1~ u a .e . y , f ( y ) = 0 ;

i f f f o r u a .e . u, ~u a .e . x and ~u a .e . z, f ( x - l z ) = 0 ;

i f f f o r u a .e . u, xu a .e . x and ~u a .e . z, f ( z - l x ) = O,

by F u b i n i ' s theorem ;

i f f f o r u a .e . u, ~u a .e . x and ~d(x) a .e . Y, f ( y - 1 ) = 0 ;

i f f f o r v a .e . x and ~d(x) a .e . y , f ( y - 1 ) = 0 ;

i f f [ v ] - l ( f ) = O.

I f u i s q u a s i - i n v a r i a n t , ~ i s a pseudo-image by d o f v - l ~

Let ~ be a measure on G O . Then a measure [~]

a s a t u r a t i o n o f u.

as above is c a l l e d

3.8. P ropos i t i on : Let mu be the s a t u r a t i o n o f the po in t mass a t u, t ha t i s , a

pseudo-image o f ~u. Then

( i ) the c lass o f mu depends on ly on the o r b i t [u] ;

( i i ) mu is ergod ic ; and

( i i i ) every q u a s i - i n v a r i a n t measure ca r r i ed by [u] is e q u i v a l e n t to ~u'

Proof :

( i ) Let N be a subset o f [u] and v be in [u] . Since ~u = x .~v f o r

x c G u d -1 (N) is ~ U - n e g l i g i b l e i f f i t is ~ V - n e g l i g i b l e . v '

( i i ) The e r g o d i c i t y o f a t r a n s i t i v e q u a s i - i n v a r i a n t measure is we l l known

(e.g. [61 ] , theorem 4 .6 , p. 278). Suppose t ha t A is a lmost i n v a r i a n t and has p o s i t i v e

measure and l e t v be S ~v dmu(V ).

Then 0 = v [ d - l ( G ~ a ) n r -Z (A) ] = SA ~V[d-1(GOA)]dmu(V ). Hence, f o r some v in A,

~V [d - l (GOA) ] = 0 and by ( i ) ~u(GO~A) = O.

( i i i ) ( c f . [61] , lemma 4 .5 , p. 277). Let u be a q u a s i - i n v a r i a n t p r o b a b i l i t y

measure such tha t ~ ( [ u ] ) = I and l e t ~ be i t s induced measure on G. Then mu is a

pseudo-image o f ,~ by d :

v[d- l (A) ] : 0 i f f f o r ~ a .e . v , ~V [d - l (A ) ] = 0 ;

i f f f o r u a .e . v, ~U [d ' l (A ) ] = 0 because o f ( i ) ;

i f f au(A) = 0 •

26

But so is u because of quasi- invariance :

~[d-Z(A)] = 0 i f f ~- l [d-Z(A) ] = 0 ;

i f f for ~ a.e. v, ~v [d - l (A ) l = 0 ;

i f f ~(A) = O.

Q.E.D.

I f G is the groupoid of a t r ans i t i ve transformation group (U,S), the class of

~u is the unique invar ian t measure class on U. This case is well known (e.g. [74],

theorem 8.19, p.25).

3.9, De f in i t i on : A t r ans i t i ve measure is a quas i - invar ian t measure carr ied by an

o rb i t . Up to equivalence, there ex is ts one and only one t r ans i t i ve measure on the

o rb i t [u] ; i t w i l l be denoted ~[u]" A quas i - invar iant ergodic measure which is not

t r ans i t i ve is cal led properly ergodic. A quas i -o rb i t is an equivalence class of quasi-

invar ian t ergodic measures.

3.10. Proposit ion : Suppose that G is second countable. The modular funct ion D of

the t r ans i t i ve measure ~[u] can be chosen such that DIG(v ) = modular funct ion of

G(v) fo r ~ [u ]a.e.v .

Proof : This is in theorem 4.4 p. 48 of [44]. An a l ternate proof is to use a s imi la-

r i t y between the essent ia l l y t r ans i t i ve groupoid (G,~ru]) and the group G(u)

(cf . I69], theorem 6.19).

3.11. A well known theorem of J. Glimm [36] states that , for a second contable l oca l l y

compact transformation group G, the fo l lowing properties are equivalent :

( i ) every o rb i t is l oca l l y closed ;

( i i ) the o rb i t space GO/G with the quot ient topology is T O ;

( i i i ) every quas i -orb i t is t r ans i t i ve .

We do not know i f th is can be generalized to a rb i t ra ry second countable l oca l l y com-

pact groupoids with Haar system. The impl icat ions ( i ) ~ > ( i i ) : > ( i i i ) may be

obtained as in [36].

27

3.12. De f in i t i on : An invar ian t measure is a quas i - invar ian t measure whose modular

funct ion is equal to 1.

3.13. Def in i t ion : ([55] p. 448). Suppose G p r inc ipa l . A quas i -o rb i t is cal led

( i ) type I i f i t is t r ans i t i ve ,

( i i ) type 111 i f i t is properly ergodic and contains a f i n i t e invar ian t measure,

( i i i ) type I I i f i t is properly ergodic and contains an i n f i n i t e invar ian t

measure, and

( iv ) type I I I i f i t is properly ergodic and contains no invar iant measure.

3.14, Def in i t ion : A pr inc ipal groupoid is of type I i f i t has type I quasi -orb i ts

only.

The notion of invar ian t measure can be extended as fo l lows. Before g iv ing the

de f i n i t i on , recal l that ZI(G, IR) is the group of continuous homomorphisms of G into IR.

Let c be in ZI(G, IR), then we denote by Min(c) the set of u's in G O such that C(Gu)iS

in[O,~) and by Max(c) the set Min ( -c ) .

3.15, De f in i t i on : Let c ~ ZI(G, IR) and B ~ [-~, +~]. We say that a measure u on

G O sa t i s f ies the (c,B) KMS condi t ion i f

( i ) when ~ is f i n i t e , u is quas i - invar iant and i t s modular funct ion D is equal

to e -Bc .

( i i ) when ~ = ± ~, the support of ~ is contained in Min (± c). A (c,~) KMS

probab i l i t y measure is also cal led a ground state for c. The point mass at u is cal led

a physical ground state i f ~in(c) n [u] = {u}.

The terminology w i l l be j u s t i f i e d in the section 4 of the second chapter.

However the condit ion D = e -~c is closer to the classical Gibbs Ansatz for equi l ibr ium

states than to the ana ly t i c form of the KMS condit ions (cf . example 3.1.6).

3.16. Proposit ion :

( i ) Note f i r s t that c ' l ( o ) is a l oca l l y compact groupoid. I f G is r -d iscre te

with Haar system, then so is c-1(0).

( i i ) Suppose that G is r -d iscre te and that B is f i n i t e . Then, a (c,B) KMS

measure for G is an invar ian t measure for c-1(0).

28

( i i i ) The subset Min(c) is closed in G O .

( i v ) The subset Min(c) is i n va r i an t under c -1 (0 ) , t ha t i s , i f x ~ c-1(0)

and d(x) c H in (c ) , then r ( x ) c Min(c) .

(v) The reduct ion of G to Min(c) is equal to the reduct ion of c-1(0) to Min(c) .

Proof : Assert ions ( i ) and ( i i ) are c lear .

( i i i ) I f u ~ Min(c) , there ex is ts x E G such that d(x) = u and c(x) < O. Let V

be an open neighborhood o f c such tha t c(y) < 0 fo r y ~ V. Then d(v) is an open

neighborhood of u and d(V) n Min(c) = @ .

( i v ) Let x E c-~1(0) w i th d(x) ~ N in (c ) . For any y c Gr(x) , yx E Gd(x) and

c(y) = c(y) + c(x) = c(yx) > O. This shows tha t r ( x ) c Min(c) .

(v) I f d(x) ~ Min(c) , c(x) > 0 and i f r ( x ) ~ Min(c) , - c (x ) = c(x) _> O, hence

c ( x ) = O.

Q.E.D.

3.17. Propos i t ion : (c f . [65] , theorem 7.5, page 26) A l i m i t po in t (w i th respect to

the vague convergence of measures) of (c,~) KMS measures when B ÷ ~ is a (c,~) KMS

measure.

Proof : Suppose tha t ~B tends to u as B tends to ~ and suppose tha t the modular

func t ion of ~ is e -Bc. Let vB be the induced measure and l e t v be the induced mea-

-1 tends to - 1 as ~ tends to ~. Therefore, sure of u. Then v~ tends to v and vB

fo r every non-negat ive f in Cc(G ), f fcd~ -1 = l im f fcd~B- I

= l im f f ce ~c dv~

= l im ( fc > 0 fce~C d ~ + Jc < 0 fceBc d~B)"

Since ce ~c tends to 0 un i fo rmly on c < O, the second in tegra l tends to O. Hence

f f cd ~I is non-negat ive fo r every non-negat ive f c Cc(G ).

Thus, c is non-negative on the support of v -1, which is d - l (suppu) . That i s , suppu is

contained in Min(c) .

Q.E.D.

The l as t par t of t h i s sect ion is devoted to the study of the r e l a t i o n s h i p between

the not ion of quas i - invar iance given in 3.2. and the usual not ion of quas i - invar iance

2g

under an inverse semi-group of t rans fo rmat ions .

Let us f i r s t look a t the case o f a t rans fo rmat ion group (U,S). Let G = U x S

be the associated groupoid. The measure on G induced by the measure ~ on U is v =

x ~, where ~ is a l e f t Haar measure of S. With respect to the groupoid G, the group

S acts in two d i f f e r e n t ways :

( i ) The ho r i zon ta l ac t ion is the ac t ion of S on U. One says tha t ~ is quasi -

i n v a r i a n t i f i t is q u a s i - i n v a r i a n t under t h i s a c t i o n , t ha t i s , ~ ~ ~-s f o r any s E S.

( i i ) The v e r t i c a l ac t ion is the ac t ion of S on i t s e l f , o r r a the r on each f i b e r

{u) x S. One notes tha t ~ is q u a s i - i n v a r i a n t under t h i s ac t ion . I f we l e t S act on the

r i g h t , dx -s - I is equal to ~(s ) , where a is the modular func t ion of S. d~

Before studying the general case, l e t us e s t a b l i s h some conventions : Let (X,u)

and (Y,v) be two measure spaces and s : X ÷ Y a bimeasurable b i j e c t i o n from X onto Y.

The image o f x by s is w r i t t e n x - s and the image of ~ by s is w r i t t e n ~- s. Thus,

~ f (y )d (~ • s ) (y ) = ~ f (x • s)du(x) f o r f E Cc(Y ). I f u " s is abso lu te l y continuous w i th

respect to v, d~ .s denotes the Radon-Nikodym d e r i v a t i v e of u" s w i th respect to v. dv

One says t ha t s is non-s ingu la r i f i t induces an isomorphism of the measure a lgebras.

3.18. D e f i n i t i o n : Let G be a l o c a l l y compact groupoid w i th Haar system {~u}. Let

be a measure on G O , not necessar i l y q u a s i - i n v a r i a n t , and v be i t s induced measure.

Let s be a G-set measurable w i th respect to the complet ion of v.

( i ) We say tha t ~ is ~ u a s i - i n v a r i a n t under s (or s is non-s ingu lar w i th respect

to v ) i f the map from ( d - l [ d ( s ) ] , V l d _ l [ d ( s ) ] ) to ( d ' l [ r ( s ) ] , V ld_Z [ r (s ) ] ) def ined by

the ru le x ~ xs - I is non s ingu la r . -1

d~s The Radon-Nikodym d e r i v a t i v e T ( w h e r e we w r i t e ~ instead of the approp r ia te

r e s t r i c t i o n ) w i l l be denoted by ~( • ,s) and ca l l ed the v e r t i c a l Radon-Nikodym

d e r i v a t i v e of s (w i th respect to v) .

( i i ) We say tha t ~ is q u a s i - i n v a r i a n t under s i f the map from ( d ( s ) , u l d ( s ) ) to

( r ( s ) , ~ i r ( s ) ) def ined by the ru le u ~ u • s -1 is non-s ingu la r . The Radon-Nikod~1 - I

d e r i v a t i v e d~. s w i l l be denoted by A ( - , s ) ca l l ed the hor i zon ta l Radon-Nikodym

d e r i v a t i v e o f s (w i th respect to ~) .

30

Remark : Since we assume that G is second countable, (G,v) is a standard measure

space. Therefore, i f s is a measurable G-set, r (s ) is measurable and the map from

r (s ) to s sending u to us is measurable.

3.19.Proposi t ion : With the notat ions of the previous d e f i n i t i o n , assume that u is

quas i - invar ian t . Then the ve r t i ca l Radon-Nikodym de r i va t i ve of a non-singular measu-

rable G-set s with respect to v depends on d(x) only. More prec ise ly , there ex is ts a

funct ion u ~ 6(u,s) defined on r ( s ) , pos i t i ve and measurable and which we s t i l l ca l l

the ve r t i ca l Radon-Nikodym de r i va t i ve of s, such that

~(d(x) ,s ) = d~s ' l dv (x) fo r v a.e. x in d - Z [ r ( s ) ] .

Proof : Let a(x) = d(vs-1) (x) be the ve r t i ca l Radon-Nikodym de r i va t i ve of s with - d ~

respect to ~. Since vs " I = fd (s ) (~Us-1) dr(u) and vs - I = f d ( s ) ~ ( ~ u ) d~(u) are two

r-decompositions of vs -1, there ex is ts a ~-conull set U in G O such that fo r every u

in U, ~u s-1 = ~xu. That i s , fo r u in U and ~u a.e. x in d - l [ r ( s ) ] , ~(x) = d(~Us-1)(x). d~ u

The commutativi ty of l e f t and r i gh t m u l t i p l i c a t i o n allows us to w r i t e , fo r any x in G U

and any pos i t i ve measurable f ,

f f ( y ) ~ ( x y ) d~d(X)(y) = f f ( x - l y ) { ( y ) d~r(X)(y)

= ~ f ( x - l y s -1) d~r(X)(y)

= ~f(ys -1) dxd(X)(y)

= f f ( y ) ~(Y) dxd(X)(y) .

Hence, for any x in G U and ~d(x) a.e. y , ~(xy) = ~(y) . Therefore, i f ¢ is a pos i t i ve

measurable funct ion such that f@d~u= 1 for u in U, the funct ion 6 defined in r (s ) by

~(u) : f~(X) ~(X) dZu(X )

has the required property. Indeed, since U is ~-conull, d-l(u) is v-l-conull and

r-l(u) is v-conull and since ~ is quasi-invariant, G U = d-l(u) m r-l(u) is v-l-conull,

hence ~u-COnull for ~ a.e.u. Thus, for u a.e. u and any positive measurable f,

~f(y) 6 od(y) d~U(y) = ~f(y) ~(x) #Ix) d~d(y)(X) d~U(Y),

= #f (y) ~(xy) O(xy) dZu(X) d~U(y),

= l (~ f (Y) ~(xy) #(xy) dzU(y)) dZu(X),

= #f(y) ~(Y) (#@(xy)dZu(X)) dxU(Y),

= #f(y) #(y) dZU(y) ; therefore

31

I f ( y ) 6od(y) dv(y) = ~ f (y ) ~(y) dv(y) ,

= I f ( ys - I ) dv(y) .

Q.E.D.

3 .20.Propos i t ion : Let u be a quas i - i nva r ian t measure, v i t s induced measure and s a

measurable G-set. Then the fo l lowing proper t ies are equiva lent :

( i ) u is quas i - invar ian t under s.

( i i ) u is quas i - invar ian t under s. Moreover, i f these condi t ions are s a t i s f i e d ,

the ve r t i ca l and the hor izonta l Radon-Nikodym der iva t i ves of s with respect to u,

~ ( . , s ) and A ( - , s ) , are re la ted by the equation ~(u,s) = D(us) A(u,s) fo r ~ a.e. u

in r ( s ) , where D is the modular funct ion of ~.

Proof : Suppose that ( i ) holds. Given a non-negative measurable funct ion h def ined

on r ( s ) , there ex is ts a non-negative measurable funct ion h def ined on d - l [ r ( s ) ]

such that h(u) = I f ( x ) d~u(X ) fo r u e r ( s ) ( c f 2.3). Then,

lh(u .s - I ) d~(u) = I f ( x ) d~ -1 (x) d~(u) u . s

= i f ( x s " I ) d~ u (x) d~(u)(by r i gh t invar iance of {~u })

= I f ( x s -1) D- l (x ) d~(x)

= If(x) D-l(xs) a(d(x),s) d~(x) = If(x) D- l (d (x )s ) ~ (d(x ) ,s ) o-Z(x) dv(x)

: I f ( x ) D - I (us) 6(u,s) d~u(X ) d~(u)

= [h(u) D-I (us) a(u,s) d~(u).

Hence u is quas i - i nva r ian t under s and d(~s-1) du (u) = D- l (us)~(u ,s ) fo r ~ a.e. u in

r(s).

Conversely, suppose that ( i i ) holds. Then, f o r any non-negative measurable func-

t ion f def ined on d - l [ r ( s ) ] ,

I f ( x s -1) dv(x) = I f ( x s -1) D(x) d~u(X ) du(u)

= I f ( x ) D(xs) d~ _l(X) d~(u)Iby r i gh t invar iance of {~u}~ us

= I f ( x ) D(xs) d~u(X ) A(u,s)dp(u)

= If(x) D(d(x)s) A(d(x ) ,s ) D(x) dv-Z(x)

= I f ( x ) D (d (x )s ) A ( d ( x ) , s ) d~(x ) .

32

This shows tha t v is q u a s i - i n v a r i a n t under s and tha t -1 dvs (x) = D(d(x)s) A (d (x ) , s ) f o r v a . e , x in d - l [ r ( s ) ] .

Q.E.D.

3.21,Case of a t rans fo rmat ion group.

Let us look back to the case of a t rans fo rmat ion group (U,S). With above no ta t i on ,

G = U x S and ~u = 5u x ~ where ~ is a l e f t Haar measure of S. For any measure u on U

and any G-set s = {u,u • s) : u E U} where s is an element of S, the induced measure

v = u x ~ is q u a s i - i n v a r i a n t under s and ~(u,s) = ~(s) where ~ is the modular func t ion

of S. I t is known ( e . g . [ 6 1 ] , theorem 4.3, page 276) tha t ~ is q u a s i - i n v a r i a n t in the

sense of 3.2 i f f i t is q u a s i - i n v a r i a n t under the group S. The hor i zon ta l Radon-Nikodym

d e r i v a t i v e A(u,s) is the usual Radon-Nikodym cocycle of the ac t ion . I f ~ is quasi -

i n v a r i a n t , i t f o l l ows from 3.20 tha t i t s modular func t ion is

D ( u , s ) : ~(s)/A(u,s).

3.22.Case o f an r - d i s c r e t e groupoid.

Since the count ing measure ~u is i n v a r i a n t under any G-set s, the v e r t i c a l

Radon-Nikod3an 6(u,s) is i d e n t i c a l l y equal to 1, independent ly of any measure u on G O .

Suppose tha t G admits a cover of compact open G-sets and l e t ~be i t s ample semi-group

( d e f i n i t i o n 2 .10) . Then a measure u on G O is q u a s i - i n v a r i a n t i f f i t is q u a s i - i n v a r i a n t

under ~ . Indeed i f ~ is q u a s i - i n v a r i a n t , by 3.20 any compact open G-set leaves

q u a s i - i n v a r i a n t . Conversely, i f ~ is q u a s i - i n v a r i a n t under 3 , i t is q u a s i - i n v a r i a n t

s ince any compact set can be covered by f i n i t e l y many compact open G-sets.

3.23.Case o f a p r i n c i p a l and t r a n s i t i v e groupoid.

Let X be a l o c a l l y compact space. As in 1 .2 .c , the graph X x X of the t r a n s i t i v e

equivalence r e l a t i o n on X ( tha t i s , any two elements of X are equ iva len t ) has a

s t ruc tu re of groupoid. With the product topo logy , i t is a l o c a l l y compact groupoid.

As in 2 .5 .c , any measure

on Xw i t h support equal to X def ines a Haar system on X x X. The t r a n s i t i v e measure

X induces the product measure m x m. A measurable G-set s is non-s ingu la r

w i th respect t o m x m i f f i t is the graph of a non-s ingu la r t rans fo rmat ion of

(X,m). The hor i zon ta l and v e r t i c a l Radon-Nikodym d e r i v a t i v e s o f s w i th respect to

33

are equal : 1

A(x,s) = ~(X,S) = d~s-~ (x) fo r ~ a.e. x in r (s) d~

The measure m is i nva r i an t because i t s modular func t ion is i d e n t i c a l l y equal to 1.

We have def ined in 3.18 the not ion of a non-s ingular measurable G-set wi th

respect to the induced measure ~ o f a measure u on G O . I t w i l l be useful to have a

d e f i n i t i o n depending only on the groupoid G an the Haar system {~u}.

3 .24, D e f i n i t i o n : Let G be a l o c a l l y compact groupoid.

( i ) A G-set s w i l l be ca l led a Borel G-set [resp. a continuous G-set I i f the

r e s t r i c t i o n of each of the maps r and d to s is a Borel isomorphism onto a Borel

subset of G O [resp. a homeomorphism onto an open subset of GO].

( i i ) Suppose that G has a Haar system {xu}. A non-s ingular Borel G-set [resp.

non-s ingular cont inuous G-set] is a Borel G-set [resp. a continuous G-set] such that

there ex is ts a Borel [resp. continuous] pos i t i ve funct ion on r (s ) bounded above

and below on compact sets, denoted ~( - ,s ) and ca l led the ve r t i ca l Radon-Nikodym

d e r i v a t i v e of s, such that

~ (d(x ) ,s ) = d~Us--~l (x) fo r every u E G O and ~u a.e. x c d - 1 [ r ( s ) ] . d~ u

Thus, a non-s ingular Borel G-set s is non s ingu lar wi th respect to the induced measure

v of every measure ~ on G O and

dvs -1 (x) fo r v a.e. x c d - l [ r ( s ) ] . ~ ( d ( x ) , s ) = d~

3.25. Examples :_ In the case of a t ransformat ion group (U,S), the G-set s =

{ (u,u • s) : u~V} where V is an open subset of U and s E S, is a non-s ingular

continuous G-set. I ts v e r t i c a l Radon-Nikodym d e r i v a t i v e is ~(u,s) = 6(s) fo r u c V,

where ~(s) the modular func t ion of S evaluated at s. In the case of a r - d i sc re te

groupoid, any open G-set s is a non-s ingular continuous G-set. We have a l ready obser-

ved that i t s ve r t i ca l Radon-Nikodym d e r i v a t i v e 5(u,s) is equal to 1, fo r u c r ( s ) .

3.26. The set o f non-s ingular Borel G-sets [ resp.non-s ingu lar continuous G-sets] is

an inverse semi-group under the operat ions ( s , t ) ÷ st and s ÷ s -1. We ca l l i t the Borel

ample semi-group o f G and denote i t ~b [resp. the continuous ample semi-group of G

and wr i te ~c ] . Let us note the fo l l ow ing formulas : fo r s , t e ~ b J

34

~ ( u , s t ) = 6 (u ,s ) ~ ( u - s , t ) f o r u c r ( s t )

6 (u ,s -1) = l ~ ( u - s - l , s ) I "1 f o r u c d(s)

3.27. D e f i n i t i o n : Let G be a l o c a l l y compact groupoid w i t h Haar system. We w i l l

say t ha t G has s u f f i c i e n t l y many non -s ingu la r Borel G-sets i f f o r every measure u

on G O w i th induced measure v on G, every Borel set in G o f p o s i t i v e u-measure

con ta ins a n o n - s i n g u l a r Borel G-set s o f p o s i t i v e u-measure, t h a t i s , such t ha t

u ( r ( s ) ) > O.

3.28. Examples :

(a) Trans format ion group. Let u be a measure on the u n i t space U o f the t r ans -

fo rma t ion group (U,S). A Borel subset o f U x S o f p o s i t i v e u × l -measure, where I

is a l e f t Haar measure f o r S, con ta ins a rec tang le A × B w i th A,B Borel~u(A) > 0

and I (B) > O. Choose s ~ B. Then s = { ( u , s ) : u c A} is a n o n - s i n g u l a r Borel G-set

o f p o s i t i v e u-measure.

(b) r - d i s c r e t e groupo ids . Let u be a measure on the u n i t space o f a second

countab le r - d i s c r e t e groupoid G. Let E be a Borel set in G o f p o s i t i v e v-measure.

Since G can be covered by coun tab ly many open G-sets , t he re e x i s t s an open G-set t

such t h a t s = E t has p o s i t i v e ~-measure. Then, s is a n o n - s i n g u l a r Borel G-set

o f p o s i t i v e u-measure.

(c) T r a n s i t i v e p r i n c i p a l g roupo ids . Let × be a l o c a l l y compact space. We de f i ne

the t r a n s i t i v e groupoid on the space X as G = X x X, w i th the groupoid s t r u c t u r e g iven

in 1 .2 .c and the product t opo logy . We know t h a t a Haar system on G is de f ined by a

measure ~ o f suppor t ×. I f X is uncountab le and s a t i s f i e s the second axiom o f counta-

b i l i t y , and i f ~ is non-a tomic , then G has s u f f i c i e n t l y many non -s i ngu la r Borel G-sets.

This can be seen as f o l l o w s : the re is a Borel isomorphism o f X onto I n c a r r y i n g

i n t o the Lebesgue measure. Thus the problem is reduced to the case X =I~ ,

= Lebesgue measure. Then the t r a n s i t i v e groupoid i s isomorphic to the groupoid

o f the t r a n s f o r m a t i o n group ( IR, IR) where IR acts by t r a n s l a t i o n and we may

conclude by a.

35

Question : Assume that G has s u f f i c i e n t l y many non-s ingular Borel G-sets and that

is a measure on G O quas i - i nva r i an t under every non-s ingular Borel G-sets ; can we con-

clude that u is quas i - i nva r i an t ?

The existence of s u f f i c i e n t l y many non-s ingular Borel G-sets w i l l be needed in

the second chapter (theorem 2.1.21) .

4. Continuous Cocycles and Skew-Products

The asymptotic range of a continuous one-cocycle ( d e f i n i t i o n 4.3) is used to

solve a few problems concerning the t r i v i a l i t y of cocycles and the i r r e d u c i b i l i t y of

skew-products. This sect ion c lose ly fo l lows [56], [57] and [58] where a s im i l a r study

has been done for C* -a lgeb ras .

Let G be a topo log ica l groupoid ( d e f i n i t i o n 2.1) . I f E is a subset o f the un i t

space G O , [El w i l l denote i t s sa tura t ion : [E] = r [ d - l ( E ) ] . I f E = [El , we say that

E is i nva r i an t (or i nva r i an t under G i f there is any ambigui ty) . We w i l l always assume

that the range map r : G ÷ G O is open. Recall (2.4) that l o c a l l y compact groupoids wi th

a l e f t Haar system have th is property. Then, the sa tu ra t ion of an open subset of G O

is open.

4.1. D e f i n i t i o n : Let G be a topo log ica l groupoid wi th open range map.

( i ) G is minimal i f the only open i nva r i an t subsets of G O are the empty set

and G O i t s e l f .

( i i ) G is i r r educ ib l e i f every non-empty i nva r i an t open subset of G O is dense.

I f there ex is ts a dense o r b i t , then G is i r r educ ib l e . The converse holds i f G is

second countable and l o c a l l y compact. I t is useful to note that the i r r e d u c i b i l i t y

of G may be expressed as the densi ty of the image of G in G O x G O by the map ( r ,d) :

G ÷ G O x G~ x ÷ ( r ( x ) , d ( x ) ) . These notions of m in ima l i t y a n d i r ~ d u c i b i l i t y could

have been def ined in terms of the s t ruc ture space GO//G of G, obtained from the quo t ien t

space GO/G by i d e n t i f y i n g o rb i t s wi th the same c losure, but we w i l l not make use of i t

36

here. The next p ropos i t ion shows tha t they are i nva r i an t under continuous s i m i l a r i t y .

4.2. Propos i t ion : Suppose tha t G and H are topo log ica l groupoids which are cont inuous-

l y s i m i l a r , t ha t i s , which are s im i l a r as in d e f i n i t i o n 1.3 where the homomorphisms

: G ÷ H and ~ : H ÷ G are cont inuous. Then the map 0 ÷ (~0 ) - I (0 ) sets up a b i j e c t i o n

between the i n v a r i a n t open subsets of H and G.

Proof : Let 0 be an i nva r i an t open subset o f H. The (@0)-1(0) is open and i nva r i an t .

For, if x ~ G and @Old(x)] c O, then @O[r(x)] : r [#O(x) ] ~ 0 since 0 is i nva r i an t .

Moreover, (90° ¢0)-1(0) = 0

Indeed (~ o@)(x) = (e o r ) ( x ) x (eod(x)) - I

therefore

@0 o@O(u) = r [ 0 (u ) ] w i th d ie (u ) ] = u

u ~ 0 i f f ~0 o #0 (u) c O.

Q.E.D.

Let G be a topo log ica l groupoid and A a topo log ica l group, not necessar i ly abe-

l i an . We may s t i l l def ine (c f 1.11) the fo l l ow ing objects. The set of continuous

homomorphisms from G to A is denoted by ZI(G,A). The subset of ZI(G,A) cons is t ing

of elements of the form c(x) = [ bo r ( x ) ] [ bod (x ) ] -1 where b is a cont inuous func t i on

from G O to A is denoted by BI(G,A). Noreover, we say tha t two elements c and c' in

ZI(G,A) are cohomologous i f there ex is ts a cont inuous func t ion b from G O to A such

tha t c ' ( x ) : [ b o r ( x ) ] c(x) [bod(x)] -1.

The fo l l ow ing d e f i n i t i o n of the asymptotic range of a cocycle is the topo log ica l

vers ion of the d e f i n i t i o n 8.2 of [31,1] .

4.3. D e f i n i t i o n : Let G be a topo log ica l groupoid, A a topo log ica l group and c an

element o f ZI(G,A).

( i ) The range of c is R(c) = c losure of c(G).

( i i ) The asymptotic range of c is R (c) = n R ( c u ) , where the i n te r sec t i on is

taken over a l l non-empty open subsets U of G O and c U denotes the r e s t r i c t i o n of c to

GIU. Moreover, l e t u be a un i t of G.

( i i i ) The range of c at u is RU(c) = c losure of c(GU).

( i v ) The asymptotic range of c at u is R u~ = n R u (Cu), where the i n t e r sec t i on

37

is taken over a base of neighborhoods o f u.

We use in the f o l l ow ing d e f i n i t i o n the character group A o f a topo log ica l group

A ; i t is the group o f continuous homomorphisms of A in to the c i r c l e g roupT .

4.4. D e f i n i t i o n : Let G be a topo log ica l groupoid, A a topo log ica l group and c an

element of ZI(G,A). The T-set of c is T(c) = {x E A : xoC E BI(G,~)} .

The fo l l ow ing p ropos i t ion gives some basic proper t ies of the quan t i t i es R (c)

and T(c) ; in p a r t i c u l a r , they depend only on the cohomology class o f c. The aim of

th is sect ion is to show t h e i r usefulness, j u s t i f y i n g t h e i r i n t roduc t ion . Further

references to the asymptotic range and the T-set o f a cocycle can be found in [31] in

the context of ergodic theory. I t is i n te res t i ng to note that they were f i r s t i n t r o -

duced on a work about operator algebras, namely, the Araki-Woods c l a s s i f i c a t i o n of

factors obtained as i n f i n i t e tensor products of factors o f type I .

4.5. Proposi t ion : Let G be a topo log ica l groupoid with open range map, A a topo lo-

g ica l group and c ~ ZI(G,A). Then

( i ) R (c) is a closed subgroup of A, T(c) is a subgroup of A, and R (c) and

T(c) are orthogonal to each other .

( i i ) R (c) and T(c) depend only on the class o f c.

( i i i ) R (e) = {e} and T(e) = A, where e denotes the i d e n t i t y element of A as wel l

as the constant cocycle e(x) = e.

Proof :

( i ) Let us f i r s t show tha t R(c) R (c) c R(c). Suppose acR(c ) and b e R (c).

For every neighborhood V of b, r [ c - l ( v ) ] is dense in G O : i f not , there would ex i s t a

non-empty open subset 0 avoiding r [ c - l ( v ) ] and co - l ( v ) would be empty. Let W be a

neighborhood of ab and choose U,V open neighborhoods of a and b respec t i ve ly such tha t

UV c W. Since d [ c ' l ( u ) ] is a non-empty open set and r [ c - l ( v ) ] is dense, there ex i s t

x, y ~ G such tha t c(x) c U, c(y) E V and d(x) = r ( y ) . Then, c(xy) = c (x )c (y )~ UV cW.

This shows a b e R(c). We deduce that R (c) is s tab le under m u l t i p l i c a t i o n : f o r any

non-empty open set U of G O , R (c) R (c) c R(Cu) R~(Cu) c R(Cu) hence R~(c) R (c) c

R (c). As i t is c losed, symmetric and contains e, R~(c) is a closed subgroup o f A.

38

Since BI(G,T), wi th pointwise m u l t i p l i c a t i o n , is a group, T(c) is a subgroup of A.

We f i n a l l y have to show that for every x ~ T ( c ) and every a ~ R (c) , x(a) = 1. For

every closed neighborhood V of I in T, there ex is ts a non-empty open set U in G O such

that (×oc)(Gu) c V because xoc ~ BI(G,T) ; in p a r t i c u l a r , x(a) ~ V.

( i i ) Suppose that c ' ( x ) = [ bo r ( x ) ] c ( x ) [bod(x)] -1 wi th c c ZI(G,A) and b a

continuous map from G O to A. Let a c RSc ). Vie want to show tha t a ~ R (c ' ) ; t ha t

i s , given a non-empty open set U' on G O and a neighborhood W' of a, we want to show

tha t W' n c ' (GIu, ) # @. We choose u E U ' , a neighborhood V of b(u) and a neighborhood

W o f a such that VWV-1c W'. There ex is ts an open neighborhood U of u such that b(U)

c V . Since W nc (G iu ) # 9, we are done. We have shown R (c) c R ( c ' ) , hence R (c) =

R ( c ' ) . The equa l i t y T(c) = T(c ' ) resu l ts from the d e f i n i t i o n o f a T-set.

( i i i ) Clear.

Q.E.D.

S imi la r proofs y i e l d s im i l a r resu l ts about the asymptotic range of a cocycle

a un i t u.

G o 4.6. Proposi t ion : Let G, A, c be as before and u ~ . Then

( i ) RU(c) R~(c) : RU(c).

( i i ) R~(c) is a closed subsemi-group o f A

( i i i ) R~(c) depends only on the class of c.

( i v ) R~(e) = {e}

(v) I f u ~ v, RU(c) = RV(c).

at

To proceed f u r t h e r , an add i t i ona l assumption on the topo log ica l groupoid G w i l l

be needed. Let us reca l l the d e f i n i t i o n 3 .24. i ;

4.7. D e f i n i t i o n : Let G be a topo log ica l groupoid. A G-set s ( d e f i n i t i o n 1.10) w i l l

be ca l led a continuous G-set i f the r e s t r i c t i o n o f r and d to s is a homeomorphism

onto an open subset o f G O .

39

An open G-set o f an r - d i s c r e t e l o c a l l y compact groupoid w i t h Haar measure i s a

cont inuous G-set . For another example, cons ider the groupoid o f a t o p o l o g i c a l t r ans -

f o rma t i on group (U,S) ; l e t V be an open subset o f U and s c S ; then the G-set s =

{ ( u , s ) : u c V} i s a cont inuous G-set . In both examples, the groupoid admits a cover

o f cont inuous G-sets. This is the assumption we need.

4 .8 . P r o p o s i t i o n : Let G be a t o p o l o g i c a l g roupo id , A a t o p o l o g i c a l a b e l i a n group and

c c Z I (G,A) .

( i ) I f c c BI (G,A) , then f o r any neighborhood V o f e in A and any u c G O , the re

e x i s t s an open neighborhood U o f u such t h a t R(Cu) c V.

( i i ) I f G admits a cover o f cont inuous G-sets , i f G O is compact and i f t he re

e x i s t s a dense o r b i t , then the converse holds.

Proof :

( i ) C lear s ince c (x ) = b o r ( x ) - b o d ( x ) .

( i i ) We assume t ha t c s a t i s f i e s the c o n d i t i o n t h a t f o r any neighborhood V of e

in A and any u c G O , t he re e x i s t s an open neighborhood o f u such t h a t c(Giu ) c V.

This means in p a r t i c u l a r t h a t c vanishes on the i s o t r o p y group bundle o f G. Let us

in t roduce the p r i n c i p a l groupoid assoc ia ted w i th G. I t is determined by the equ iva-

lence r e l a t i o n ~ on G O . As a se t , i t is the image o f the map ( r , d ) : G -~ G O x G O .

We p rov ide i t w i th the f i n a l t opo logy , which is u s u a l l y s t r i c t l y f i n e r than the t o p o l o -

gy induced from G O x G O . The cocyc le c f ac to r s through the map ( r , d ) :

c (x ) = c ' ( r ( x ) , d ( x ) ) . Let V be a neighborhood o f 0 in A. Since {U non-empty open set

in G O such t h a t c ' (UxU) V} is an open cover o f G O , t he re e x i s t s a f i n i t e subcover

hence an entourageCLbof the u n i f o r m i t y on G O such t h a t

(u ,v ) cqJoand u ~ v ---->c'(u,v) c V.

Let us show t h a t c' is cont inuous w i th resoec t to the t opo logy induced from G O x G O .

Given (u , v ) c G O x G O w i th u m v , l e t x c G be such t ha t r ( x ) = u and d(x) = v and

l e t s be a cont inuous G-set con ta in i ng x. Consider ( u ' , v ' ) w i th u' m v ' . For u'

s u f f i c i e n t l y c lose to u, t he re e x i s t s y c s w i t h r ( y ) = u' and c' ( u ' ,w ) - c ' ( u , v ) ,

where w = d ( y ) , can be made a r b i t r a r i l y smal l . On the o the r hand c ' ( u ' , v ' ) - c ' ( u ' , w )

= c ' ( w , v ' ) can be made a r b i t r a r i l y sma l l , p rov ided t ha t v ' is c lose enough

40

• " V to w, t h i s happens i f u' is su f f~c len t l~ c lose to u and v' s u f f i c i e n t l y c lose to v.

Next we show tha t c' is un i fo rm ly continuous on the dense subset [Uo] x [Uo] of

~0 x G O , where u 0 has a dense o r b i t . I f u, v, u ' , v ' are in the o r b i t o f u O, then

c ' ( u ' , v ' ) - c ' ( u , v ) = c ' ( u ' , u ) - c ' ( v ' , v ) . Therefore, c' extends to a continuous

on G O x G O . Then, f ( u ) = c '(u,UO) is a continuous func t ion on G O func t ion and c t

wi th i t s coboundary on [u O] x [Uo],hence on G. agrees

4.9. Propos i t ion : Let G be a t opo log i ca l groupoid admi t t i ng a cover of continuous

G-sets, A a t opo log i ca l abe l ian group and c E ZI(G,A) . Suppose u 0 c G O has a dense

o r b i t • Then R~(c) = R(Cu), where the i n t e r s e c t i o n is taken over a base o f neigh-

borhoods of u O.

Proof : Suppose tha t a c R(Cu) f o r every U in a base of neighborhoods of u O. Let

V,W be neighborhood of e on A such tha t W + W c V and U be a non-empty open set .

There ex i s t s x ~ G wi th r ( x ) = u O, d(x) c U and a continuous G-set s conta in ing x.

We may assume tha t d(s) c U and c(s) - c(s) c W.Because a c R(Cr(s) ) , there ex is ts

y ~ Gl r (s ) such tha t c(y) c a + !~• Let z = s - l y s , then z ~ Gld(s ) c GIU and

c(z) = c ( s - l r ( y ) ) + c(y) + c (d (y )s ) ~ a + W + Wc a + V•

Thus, a c R(Cu) f o r any non-empty open set U.

Q•E •D.

The f o l l o w i n g theorem may be compared w i th theorem 9 o f [31 ,1 ] . Combined w i th

the resu l t s o f the second chapter , i t y i e l d s a p a r t i c u l a r case o f a wel l -known theorem

of Sakai which s ta tes tha t every bounded d e r i v a t i v e o f a simple C*-algebra w i th

i d e n t i t y is inner .

4.10. Theorem : Let G be a t opo log i ca l groupoid admi t t i ng a cover o f continuous

G-sets and a compact un i t space, l e t A be a t opo log i ca l abe l ian group and l e t

c ~ ZI (G,A) . Assume tha t G is min imal . I f R(c) is compact and R (c) = {0 } , then

c c BI(G,A).

Proof : We use 4.8 ( i i ) . Suppose tha t there ex i s t s an open neighborhood V of 0 in

A, u e G 0, a base of neighborhoods of u and a net {x U} such t h a t

x u ~ GlU and c(x u) ~ V.

41

I f {a U} is a subset of {C(Xu)} converging to a, then a # V and a E n R(Cu) where the

i n te rsec t i on is taken over a base of neighborhoods of u. By 4.9, a c R (c). Since

R (c) = {0} , t h i s is a con t rad i c t i on . Q.E.D.

Note that i f A is to rs ion f ree , the cond i t ion R(c) compact a l ready

impl ies R (c) = {0~.

The next theorem may be compared wi th th6or6me2.3.1 of [13] in the context of von

Neumann algebras and wi th theorem 4.2 o f [56] in the context of C ~ -a lgeb ras . The

proof is adapted from [56].

4.11. Theorem : Let G be a topo log ica l groupoid admit t ing a cover of continuous G-sets

and a compact un i t space, l e t A be a l o c a l l y compact abel ian group and l e t c c ZI(G,A).

Assume that G is minimal, then i f R(c)/ R (c) is compact in A/R (c) , i t fo l lows that

T(c) is the a n n i h i l a t o r o f R (c) in A.

Lemma : Let G be a topo log ica l groupoid admit t ing a cover of continuous G-sets, l e t

A be a l o c a l l y compact abel ian group and l e t c ~ ZI(G,A). Assume that G is i r r e d u c i -

ble. Then iS= {V +R(cu) : V compact neighborhood of 0 in A and U non-empty open

subsets of G O } is a base of a f i l t e r . I ts i n te rsec t i on is R (c).

Proof : As in 3.4. of [56] , i t su f f i ces to show that given a compact neighborhood V

of 0 in A and non-empty open s u b s e t s U i of G 0, i = 1 , 2 , t h e r e e x i s t non-empty open

subsets U i c U i , i = 1,2 such that R(Cu~ )~ c V + R(Cu~ ) i , j = 1,2 and i ~ j . We choose

x c G with r ( x ) c U 1 and d(x) e U 2 and a con t inuous G-se t s c o n t a i n i n g x. We may

assume that r (s) c UI, d(s) c U 2 and c(s) - c(s) c V. Then U~ = r (s ) and U½ = d(s)

wi l l do.

Proof of the theorem : With the notat ions of the lemma, the image o f ~ in A/R (c) is

a base o f a f i l t e r of compact sets wi th i n te rsec t i on {0} . Hence, given a neighborhood

V of 0 in A, we may f ind a non-empty open set U in G O such that R(Cu) c V + R (c).

Thus, i f x is orthogonal to R (c) , R (×o c) = { I } . By 4.10, xoC ~ B I ( G , T ) , that i s ,

x e T(c). The reverse inc lus ion has been shown in 4.5. ( i ) .

42

Z 1 Recall t h a t , given a groupoid G, a group A and c ~ (G,A), one may de f ine the

skew-product G(c), whose under ly ing space is G x A and u n i t space is G O x A. I f G and

A are t opo log i ca l and c cont inuous, G(c) w i th the topology o f G x A is a t opo log i ca l

groupoid. Note tha t i f G has an open range map [ resp. a cover o f continuous G-se ts ] ,

then so has G(c).

The f o l l o w i n g c h a r a c t e r i z a t i o n of the asymptot ic range o f a cocycle in terms of

the skew-product is taken from Pedersen [60] 8 .11.8. I t w i l l be used in Section 5 of

Chapter 2. Recall t ha t there is a canonical ac t i on of A on the skew-product G(c),

given by

(x ,b ) • a = ( x , a - l b )

4.12. Propos i t i on : Let G be a topo log i ca l groupoid w i th open range map, l e t A be

a topo log i ca l abe l ian group and l e t c ~ ZI (G,A) . Then the f o l l o w i n g p rope r t i es are

equ iva len t f o r a e A :

( i ) a ~ R (c) and

( i i ) f o r any non-empty open i n v a r i a n t subset 0 of the u n i t space G(c) , 0 n 0 - a

is non-empty.

Proof :

( i ) ~ ( i i ) Suppose a E R (c ) .

Let 0 be a non-empty i n v a r i a n t subset of G O × A. I t contains a non-empty rec tang le

U × V, w i th U open on G O and V open in A. Let b ~ V. Since a c R ( c ) , t h e r e ex i s t s

x c GIU such t ha t c (x ) E a - b + V. Then ( r ( x ) , b ) and ( d ( x ) , b - a + c ( x ) ) belong to

U x V n 0. Since ( r ( x ) , b - a) is equ iva len t to ( d ( x ) , b - a + c ( x ) ) , i t belongs to 0.

Since ( r ( x ) , b - a) = ( r ( x ) , b) . a, i t a lso belongs to 0 . a .

( i i ) ~ , ( i ) Suppose tha t a s a t i s f i e s ( i i ) .

Let U be a non-empty open set in G O and V be a neighborhood of 0 in A. Choose a neigh-

borhood N of 0 such tha t W - W c V. Since the sa tu ra t i on of U x W in the un i t space

o f G(c) is an i n v a r i a n t open se t , i t contains an element (v ,b ) toge ther w i th ( v , b - a ) .

This imp l ies the ex is tence o f x and y in G such tha t

r ( x ) = v and ( d ( x ) , b + c (x ) ) e U x W and

43

r ( y ) : v and ( d ( y ) , b - a + c ( y ) ) c U x W.

Then, x - l y ~ GIU and c ( x - l y ) = -c(x) + c(y) c a + W - W c a + V.

This shows that a e R (c).

Q.E.D.

4.13. Proposi t ion : Let G be a topo log ica l groupoid with open range map, l e t A be

a topo log ica l group and l e t c ~ ZI(G,A). The fo l l ow ing proper t ies are equ iva len t :

( i ) G is i r r educ ib l e and R (c) = A and

( i i ) G(c) is i r r educ ib l e .

Proof :

( i ) ~ > ( i i ) I t su f f i ces to show tha t , given non-empty open sets U1,U 2 in G O ,

a neighborhood V of e in A and a ~ A, there ex is ts z ~ G such that r (z ) ~ U 1, d (z )c U 2

and c(z) c aV. Choose W, open neighborhood of e such that W-Iw c V. Since G is

i r r e d u c i b l e , there ex is ts b ~ A such tha t c- l (bw) n r-1(U1 ) n d - l (u2 ) # @. Let

U = r [ c - l (bw) N r - l ( U l ) n d - l ( u 2 ) ] . Since ba -1 ~ R (c) , there ex is ts x ~ GIU such

that c(x) m bWa -1. Since r (x ) E U, there ex is ts y ~ G such that r (y ) = r ( x ) ,

d(y) ~ U 2 and c(y) e bW. Let z = x - l y . Then r (z ) = d(x) ~ U c U 1, d(z) = d(y) c U 2

and c(z) = c ( x ) - l c ( y ) c a W-1W c aV.

( i i ) ~ > ( i ) I f G(c) is i r r e d u c i b l e , then G is c l e a r l y i r r educ ib l e . To show

that R (x) = A, l e t a E A, l e t V and W be neighborhoods o f e in A such that W'Iw c V

and l e t U be a non-empty open subset of G 0. Since G(c) is i r r e d u c i b l e , there ex is ts

x ~ G U and b ~ W such that bc(x) ~ Wa. Then c(x) ~W-1Wac Va. This shows that

a ~ R ( c ) .

Q.E.D.

4.14. Proposi t ion : Let G be a topo log ica l groupoid wi th open range map, l e t A be a

topo log ica l group and l e t c ~ ZI(G,A). Let (u,a) ~ G O × A.

( i )

c at u,

( i i )

I f (u~a) has a dense o r b i t r e l a t i v e to G(c), then the asymptotic range of

RE(c ) , is equal to A.

Conversely, i f G is minimal and i f R~(c) = A, then (u,a) has a dense o r b i t .

44

Proof :

( i ) Suppose tha t the o r b i t [ ( u , a ) ] = { ( d ( x ) , a c ( x ) ) : x ~ G u} is dense in

G O x A. Let b ~ A, l e t V be a neighborhood of b and l e t U be an open neighborhood

of u.There ex i s t s x e G u such t ha t ( d ( x ) , ac (x ) ) c U × aV. Thus, x e G u n GIU and

c(x) e V. We conclude tha t b c R~(c).

( i i ) Suppose tha t R~(c) = A. Let F be the c losure o f the o r b i t o f ( u , a ) . For

any b e A, (u,b) c F : indeed, l e t U be an open neighborhood o f u and V a neighbor-

hood of b ; since a - l b ~ R~(c), there ex i s t s x such tha t r ( x ) = u, d(x) E U and

c(x) m a - l v ; in o ther words, ( d ( x ) , a c ( x ) ) E U × V. The set {v c G O : f o r any b e A,

(v ,b ) e F} is non-empty, G - i nva r i an t and c losed. Since G is min imal , t h i s is G O ,

hence F = G O × A.

Q.E.D.

4.15. Propos i t ion : Let G be a t opo log i ca l groupoid w i th open range map, A a topo lo -

g ica l group and c ~ ZI (G,A) . Assume tha t A is compact,then R (c) = P,U(c) f o r every

u e G O w i th a dense o r b i t .

Proof : I#e f i r s t show tha t R(c) = RU(c)-iRU(c) f o r u w i th a dense o r b i t . The i nc l u -

sion RU(c) -1 RU(c) c R(c) holds fo r a r b i t r a r y u. Suppose now tha t a c R(c) and u has

a dense o r b i t . Since A is compact, i t su f f i ces to show tha t a belongs to the c losure

of RU(c) -1 RU(c). I f V is a neighborhood of a, r [ c - l ( v ) ] n [u] is non-empty : there

e x i s t x ,y such tha t c(x) E V, r ( x ) = d(y) and r ( y ) = u. Then, c(y) - I c(yx) ~ [c(GU) " I

c(GU)] n V. Therefore R(Cu) = RU(cu ) - I RU(cu ) f o r any open neighborhood U o f u. Using

the compactness of A, one may w r i t e :

u -1 RU(c) : R (c) R (c) = nR(cu ) = [nRU(cu)] -1 [nRU(cu )] = R (c)

where the i n t e r sec t i ons are taken over a l l open neighborhoods of u. The l a s t

e q u a l i t y holds because, in a compact group, any closed semi-group is a group.

Q.E.D.

4.16. Co ro l l a r y : Let G be a topo log i ca l groupoid w i th open range map, l e t A be a

t opo log i ca l group and l e t c m z l (G ,A ) .

( i ) I f G(c) is min imal , then G is minimal and R (c) = A

( i i ) I f A is compact, i f G is min imal , and i f R (c) = A, then G(c) is min imal .

45

Proof :

( i )

( i i )

o r b i t .

I f G(c) is minimal, G is c l e a r l y minimal. Moreover, R (c) = A by 4.12.

Using 4.14 and 4.13 ( i i ) , o n e sees tha t every (u,a) ~ G O × A has a dense

Q.E.D.

4.17. Propos i t ion : LetG be a topo log ica l groupoid with open map, l e t A be a group

71(G,A). The follow,ring propert ies are with the d iscre te topology and l e t c ~ ~

equ iva lent :

( i ) G is i r r educ ib l e and R (c) = R(c) ; and

( i i ) c - l ( e ) is i r r educ ib l e .

Proof :

( i ) ----> ( i i ) Let U 1 and U 2 be non-empty open sets in G O . By i r r e d u c i b i l i t y of

G, there is a cA such that c-1(a) n r-1(U 1) n d- l (u2 ) is non-empty. Then

U = r ~ ' 1 ( a ) n r - l ( u1 ) n d- l (u2 )] is a non-empty open set and since a "1 E R J c ) ,

there ex is ts x ~ G U wi th c(x) = a -1. Therefore, there is y e G such that d(x) = r ( y ) ,

c(y) = a and d(y) ~ U 2. Consider z = xy : d(z) ~ U 2, r (z ) = r (x ) c U 1 and

c(z) = c(x)c(y) = e. This shows that the groupoid c-1(e) is i r r educ ib l e .

( i i ) ~ > ( i ) I f c - l ( e ) is i r r e d u c i b l e , so is G.Consider a ~ R(c) and U a non-

empty open subset of G O . Since c '1 (e) is i r r e d u c i b l e , c - l ( e ] n r - l ( u ) n d - l [ r ( c - l ( a ) ) ]

= V is a non-empty open set and so is c - l ( e ) n r - l [ d ( V ) ] n d-1(U). Therefore, we can

f ind x, y, z such that : c(x) = e, c(y) = a, c(z) = e, d(x) = r ( y ) , r (z ) = d(y) ,

r (x ) m U and d(z) ~ U. Then, xyz E GIU and c(xyz) = a. This shows that a ~ R (c).

Q.E.D.

Another subgroup of A can be attached to a cocycle c e ZI(G,A) (cf . [62],

theorem of Section 2). We conclude th is sect ion by discussing b r i e f l y how i t is

re la ted to R (c) and T(c) ± (the a n n i h i l a t o r o f T(c) in A) in a p a r t i c u l a r case.

4.18. D e f i n i t i o n : Let G be a topo log ica l groupoid, l e t A be a topo log ica l group

and l e t c c ZI(G,A). We def ine R1(c ) to be the set of elements a of A wi th the

property that fo r every G(c ) - i nva r ian t complex-valued continuous funct ion on G O × A

and fo r every (u ,b) in

4.19. Propos i t i on :

46

G O × A, the e q u a l i t y f ( u ,ba ) = f ( u , b ) holds.

Let G be a topo log i ca l groupoid, l e t A be a t opo log i ca l group

and l e t c ~ ZI(G,A). Then R ( c ) c R 1 ( c ) c T(c ±

Proof : R (c) c Rl(C ). I f a ~ Rl (C) , there is a continuous func t ion f on G O × A,

which is G(c) i n v a r i a n t , and (u,b) c G O × A such tha t f ( u ,ba ) # f ( u , b ) , hence there

e x i s t an open neighborhood U of u and a neighborhood V of a such tha t f(U × bV) n

f(U x bVa - I ) = ~ . I f x E G U and c(x) c V, then f ( r ( x ) ,b ) = f ( d ( x ) , b c ( x ) ) . This is a

c o n t r a d i c t i o n and the re fo re a ~ R (c ) .

RI(C) c T(c) ±. Let a E RI(C ) and X ~ T(c ) , t ha t is xoc ~ BI(G,~) , Then, there

ex i s t s g : G O ÷# continuous such tha t god(x) ×oc(x) = gor (x ) f o r every x e G. Let

f ( u , b ) = g(u) x (b ) . Then f is continuous and G ( c ) - i n v a r i a n t . Therefore, f ( u ,ba ) =

f ( u , b ) t ha t i s , g (u )x (b )× (a ) = g ( u ) x ( b ) , hence x(a) = 1.

Q.E.D.

More in fo rmat ion can be obtained in the case of a compact abe l ian group A.

4.20. Propos i t ion : Let G be a topo log i ca l groupo id , l e t A be a topo log i ca l group

and l e t c e ZI (G,A) . Assume tha t G is minimal and A is compact and abel iano Then

RL(C) = T(c) ~.

Proof : Let f be a continuous G(c)-invariant function on @0 #A. For each X E A,

g(u) = f f ( u , a ) x(a)da is continuous and s a t i s f i e s

god(x) Xoc(x) = go t ( x )

Since G is min imal , e i t h e r g vanishes i d e n t i c a l l y o r not at a l l and, in the l a t t e r

case, XOC ~ BI (G,~ ) , t ha t i s , x ~ T(c) . Thus, f o r every u, the Four ie r t ransform of

f ( u , - ) is supported on T(c) . Hence, i f a c T(c) ~, then fo r any b c A

f ( u , a + b) = f ( u , b ) . So a c R1(c ).

Q.E.D.

We also reca l l t ha t under the hypotheses of 4.11, R (c) = R1(c ) = T(c) i . These

l a s t f ac t s , combined wi th 4.15, g ive a theorem of Rauzy ( [ 6 2 ] , theorem of sect ion 2)

about the m i n i m a l i t y of a skew-product.

CHAPTER I I

THE C* -ALGEBRA OF A GROUPOID

F i rs t , l e t us say that "groupoid" stands for l oca l l y compact groupoid

with a f ixed Haar system (de f in i t i on 1.2.2) chosen once for a l l . We shall see (corol-

lary 2.11) how the C*-algebra can be affected by another choice of Haar system. We

also assume that the topology of the groupoid is second countable.

The goal here is to construct the C*-algebra of a groupoid in a way which

extends the well-known cases of a group (e.g. Dixmier [19]) or of a transformation

group (e.g. Effros-Hahn [23]). In fact , our construction closely fel lows [23]:

the space Cc(G ) of continuous functions with compact support is made into a * -algebra

and endowed with the smallest C*-norm making i ts representations continuous ;

C*(G) is i ts completion. The deta i ls are in Section 1. We re f ra in from putt ing any

modular function in the de f i n i t i on of the invo lu t ion, since none is avai lable. However,

th is is a minor change and the C*-a lgebra so obtained is isomorphic to the usual one in

the case of a transformation group. Let us note that , in the case of a transformation

group, the * -a lgebra Cc(G ) has been studied by Dixmier ( [16],§ X) in the context of

quasi-uni tary algebras.

I f a is a continuous 2-cocycle on G with values in the c i rc le group, the

C*-a lgebra C*(G,~) is defined in the same fashion. One of the main j us t i f i ca t i ons

for i ts int roduct ion, besides the need to deal with project ive representations, is

given in Section 4, where the C*-a lgebra of an r -d iscrete pr incipal groupoid is

characterized, under sui table condit ions, by the existence of a pa r t i cu la r l y nice

kind of maximal abelian subalgebra. One of these conditions is amenabi l i ty, which is

defined in Section 3.

48

An essent ia l tool in the study of the C * -a l geb ra of a groupoid is the corres-

pondence, very f a m i l i a r in the case of a group, between the un i t a r y representat ions

of the groupoid and the non-degenerate representat ions of the C* -a l geb ra . I t is

estab l ished at the end of Section 1 and under a cond i t ion (existence of s u f f i c i e n t l y

many non-s ingu lar G-sets) s u f f i c i e n t fo r our app l i ca t i ons .

Of p a r t i c u l a r i n te res t are the regu lar representat ions of a groupoid. They have

been studied ex tens ive ly since Murray and Von Neumann and we re fe r to Hahn [45]

fo r f u r t he r d e t a i l s . They appear under var ious forms, one of them is as representat ions

induced from the un i t space ; th i s is described in Section 2, where the inducing

process from more general subgroupoids is also considered.

The l as t sec t ion , Section 5, i n te rp re t s in the language of C* -a lgebras the

resu l t s of Section 4 of the f i r s t chapter. They center around the question of p r im i -

t i v i t y and s i m p l i c i t y of a crossed-product algebra.

I . The Convolut ion Algebras Cc(G,~) and C* (G,~)

Let G be a l o c a l l y compact groupoid wi th l e f t Haar system {~u} and l e t ~ be a

continuous 2-cocycle in Z2(G,T ). For f and g c Cc(G ), l e t us def ine

f * g ( x ) = I f ( x y ) g ( y -1) ~ (xy ,y -1 )d~d(X) (y ) ,

f * ( x ) = f ( x -1) o ( x , x - l ) .

1.1. Proposi t ion : Under these operat ions, Cc(G ) becomes a topo log ica l ~ -a lgebra ,

denoted by Cc(G,~ ).

Proof : We f i r s t show tha t these operat ions are wel l def ined. For each x , f . g ( x )

is the value of the in tegra l of a cont inuous func t ion w i th compact support. Since

f * g ( x ) is nonzero only i f there is y such that f ( x y ) and g(y-1) are nonzero,

s u p p ( f . g ) is contained in the compact set (suppf)(suppg). To show the c o n t i n u i t y o f

f ~ g, we may use the same device as Connes in [14] 2.2. That i s , since G 2 is a

closed subset of the normal space G x G, the func t ion (x ,y ) -~ f ( x y ) ~ ( x y , y -1) may be

extended to a bounded continuous func t ion k on G x G. Since the func t ion

49

x ~ ~ : G ~ Cc(G),

where ~(y) = k(x,y)g(y-1), is continuous, so is the function

(x,u) ÷)k(x,y)g(y-1)d~U(y) : G x G O ÷C ; in part icular, i ts restr ic t ion to

(x,d(x)) is continuous. Note that f * is also continuous, with compact support suppf*

= (suppf) -1. The convolution is associative : i f f , g, h ~ Cc(G),

f * (g . h) (x) = ! f (xy) g h(y -1) o(xy,y-1)d~d(X)(y)

. l i f (xy ) g(y- iz) h(z -1) ~(y-lz,z-1)~(xy,y-1)d~r(Y)(z)d~d(X)(y)

= ) ) f (xy) g(y- lz) h(z - I ) {(xy,y -1) ~(y-lz,z-1)d~r(Z)(Y)dxd(X)(z)

= i ) f (xzy) g(y-1) h(z-1) {(xzy,y- lz-1)o(y- l ,z-1)d~d(Z)(y)d~d(X)(z)

= )) f (xzy g(y-1) h(z- l ) ~(xzy,y-1) ~(xz,z-1)d~d(Z)(y)d~d(X)(z)

= (f*g

= (f.g

The involution s involutive :

f**(x) = f*(x -I) ~(x,x -I

Also

(xz) h(z - I ) u(xz,z-1)dxd(X)(z)

. h ( x ) .

= f(x) ~(x- l ,x) ~(x,x - I ) = f (x) .

( f * g ) * ( x ) = f . g ( x -1) ~(x,x -1) = . ( f (x- ly) g(y-1) ~(x- ly,y-1) ~(x,x-1)d~r(X)(y).

Using ~(x- ly, y - l ) = ~(y,y-1) c~(x-l,y)

and ~(x, x- l ) = ~(x- l ,Y) ~(x- ly ,y - Ix) o (y ,y ' ] x ) , we obtain

( f . g ) * ( x ) = i g ( y - l ) { ( y , y - ! ) f (x - ly ) ~(x- ly ,y - lx ) ~(y,y ' lx)d~r(X)(y)

= I g * ( Y ) f * ( y - l x ) ~(Y'y-lx)d>r(X)(Y)

= g* * f*(x) .

Final ly, the operations are continuous. I f fn ~ f and gm -~ g' there exist

compact sets K and L such that, eventually, supp f cK and sump gm c L. Then, n

supp fn*gm c KL. Also,

If * g(x) - fn * gm (x)! ~-l]f(xy)g(y-1) - fn(xy)gm (y-1)]d~d(x)(y)

_<.l[f(xy) - fn(XY) iIg(y-i)Id~d(X)(y)

+lIfn(XY ) IIg(y - I ) -gm(y - I ) Id~,d(X)(y)

50

Therefore, fn*gm converge uni formly to f , g on KL. Moreover, supp fn a K -1 and

Ifn(X) - f * ( x ) I = Ifn(X -1) - f ( x - 1 ) I converges uniformy to zero on K - I .

q.E.D.

1.2. Proposit ion : I f a and ~' are cohomologous, then Cc(G,~ ) and Cc(G,~' ) are

isomorphic.

Proof : I f o ' ( x , y ) = a ( x , y ) c ( y ) c ( x y ) - l c ( x ) we can define the isomorphism ¢ from

Cc(G,~' ) to Cc(G,d ) which sends f to fc. Indeed for f ,g s Cc(G )

¢( f ) . ¢(g)(x) = j f ( x y ) c ( x y ) g ( y - I ) o(xy ,y-1)dxd(X)(y)

= .I f (xy)g(Y -1)

¢( f ~ g)(x)

¢ ( f ) * (x) = ¢ ( f ) ( x ' l i ~(x,x - I )

=

o ' ( x y , y ' l ) c ( x ) d x d ( X ) ( y )

= f(x - I ) c(x - I ) o(x,x -1)

f ( x - l i ~ ' ( x , x -1) c(x) = # ( f * ) ( x ) .

Q.E.D.

1.3. De f in i t i on : A representat ion of C~(G_,o) on a H i l be r t space H is a *-homomor-

phism L : Cc(G,o ) ÷~ (H) which is continuous when Cc(G,~ ) has the induct ive l i m i t

topology and~5(H) the weak operator topology, and is such that the l i nea r span of

{ L ( f ) ( , f c Cc(G,o),~ c H} is dense in H.

The I-norm introduced by P. Hahn in ~5] , page 38, w i l l be a convenient estimate

fo r the C*-norm we wish to def ine on Cc(G,a ). I t is worthwhile to not ice that th is

norm is used by numerical analysts , in the case when G is the t r i v i a l equivalence

re l a t i on on a set of n elements (e.g. in i n te rpo la t i on theory) . Let us reca l l i t s

d e f i n i t i o n - or ra ther , the d e f i n i t i o n appropriate to our se t t i ng . For f ~ Cc(G )

] I f l l l , r = sup j - I f l d ~ u, I I f i l l , d = sup I I f ld~ u ; and [Ifll I = max( IIf l l l , r , l l f I I l ,d ) - u~G 0 u~G 0

1.4. Proposit ion :

( i ) If. If I is a norm on Cc(G ) def in ing a topology coarser than the induct ive l i m i t

topology.

51

( i i ) For any oBZ2(G,T), II IT I is a * -a l geb ra norm on Cc(G,~ ).

Proof :

( i ) I t suf f ices to look at I I f I I i , r . I t is rout ine to check that i t is a norm.

÷ 0 in Then because of the con t inu i t y of the map Suppose that fn Cc(G)" '

: Cc(G) ÷ Cc(GO ) which sends f to I fd~ u, i t fo l lows that ~(IfnT) tends to zero in

Cc(GO ) and a f o r t i o r i in the space Co(GU ) of continuous funct ions tending to 0 at

i n f i n i t y , equipped with the supnorm.

( i i ) To show that Il l , gll I < l lf l l I IIgIII, i t su f f ices to consider IT I l l , r . Then

fo r f ,g c Cc(G ),

l ' I f * gl d~u < i . I I f ( y ) I Ig(y-mx)Id~r(X)(y)d~U(x) , (becauseI~ I =1)

< I I I f ( y ) l l I g ( y - l x ) I d ~ r ( y ) ( x ) d~U(y)

< i l f ( Y ) [ l ' I g (x ) Id~d(Y) (x ) d~U(Y)

< sup ~Ig(x)Id~V(x) x i l f ( y ) I d ~ U ( Y ) < I lg I I i , r l I f l i l , r-

F i na l l y , by d e f i n i t i o n , llf~l I = llf l l I .

Q.E.D.

1.5. De f in i t i on : A representat ion L of Cc(G,~ ) w i l l be cal led bounded i f

ltL(f)tE lIrllz fo r a l l f ~ Cc(G,~ ).

We may def ine Ilfll = sup IJL(f)tl where L ranges over a l l bounded representat ions

of Cc(G,~ ), and make two comments.

( i ) C lear ly I1-11 is a c ~ -semi-norm. I t w i l l be shown soon, by exh ib i t i ng

s u f f i c i e n t l y many bounded representat ions, that i t is a norm.

( i i ) For a large class of groupoids ( inc luding transformat ion groups), we w i l l

es tab l ish in Coro l lary 1.22 that every representat ion on a separable H i l be r t space is

bounded. This is done in a fashion s im i l a r to [24], 4.9, page 45 in the case of a

t ransformat ion group.

The notion of H i l be r t bundle (or more prec ise ly H i lbe r t space bundle) used in

the next d e f i n i t i o n is given in [61]page 264. The base space of such a bundle is a

standard measure space and each f i b e r is a separable H i l be r t space.

52

1.6. D e f i n i t i o n : Let o be a (continuous) 2-cocycle. A o- representa t ion o f G consists

of a quas i - i nva r i an t measure ~ on G O and a (o ,G) -H i l be r t bund le%ove r (GO,~). More

p rec ise ly , there is a map L : G + Iso(J~) = { isometr ies #u,v :J~v - ~ where u, v ~ G O }

such that

( i ) L(x) sends J~d(x) onto JCr(x) For x e G and L(u) = idJ~ 0 for u ~ G O "

0

( i i ) L (x)L(y) = ~(x ,y) L(xy) for v~ a.e. ( x , y ) , where v 2 is the induced measure

on G 2 ;

( i i i ) L(x) -1 = ~ fo r ~ a.e. x, where v is the induced measure on G ; and

( i v ) x ~ (L(x) ~od(x), ~or(x)) is measurable fo r ever'y pa i r of measurable

sections ~ and n.

Two ~-representat ions (u,JC,L) and (~ ' ,JC' ,L ' ) are equ iva lent i f the measures

and u' are equ iva lent and there ex is ts an isomorphism ~ of JContoJ£ ' ( in the sense o f

[~1]) which in ter tw ines L and L ' , that i s , such tha t L ' ( x ) ~.d(x) = ~or(x) L(x) fo r

v a . e . x .

Let (u , j~ ,L ) be a ~- representa t ion o f G. Then r (J~), or r(J~) when there is no

ambiguity about ~, denotes the H i l be r t space of square- in tegrab le sect ions wi th

respect to u. The modular funct ion of ~ is denoted by D and i t s symmetric induced

measure is ~0 = D-I /2v (see 1.3.4) .

1.7. Propos i t ion : Let (p,J~,L) be of a o- representa t ion o f G.For g,n c r (J~) and

f e Cc(G ), set

( , ) ( k ( f )~ ,q ) = ! f ( x ) (L(x) ~od(x), nor (x ) ) dvo(X ).

( i ) This def ines a bounded representa t ion of Cc(G,o ) on r(JC).

( i i ) Two equ iva lent o- representat ions of G give two equ iva lent representat ions

of Cc(G,o ) .

Before s ta r t i ng the proof , l e t us make two remarks.

a. Let Up = {u c G O : dim J£ u = p} fo r p = 1,2 . . . . . ~. I t is an i nva r i an t measurable

subset. Let Up be the r e s t r i c t i o n of u to Up.

Then (pp ,%,L) is a o - representa t ion of G, which def ines by (*) an operator Lp( f )

o n r ~ ( J £ ) ; the operator L( f ) is the d i r ec t sum of the L p ( f ) ' s . Therefore, i t is

53

su f f i c ien t to consider the case when dim J~u is constant. Then JC, is isomorphic to a

constant Hi lber t bundle with f iber K and ? (J~) is isomorphic to the space L2(GO,u,K)

of square-integrable K-valued functions on (GO,u). Moreover, for f E Cc(G ) and

~ L2(GO,~,K), L( f ) is given by

L(f)~ (u) = ~ f (x ) k(x) ~od(x) D-1/2(x) d~U(x) ~ a.e.

where the r igh t hand-side may be interpreted as a weak integral in K.

b. In the case of a group, G O is reduced to one element and there is a unique inva-

r ian t measure class. Therefore, there is no need to mention i t . Then, a ~-representa-

t ion in the sense of 1.6 is a project ive representation in the usual sense (e.g. [74]

page I00). Assume that ~ = i . Then, the representation given by ( * ) is the integra-

ted form of the unitary representation L. I t is not the usual expression since our

de f in i t i on of the involut ion d i f f e rs from the usual one by the absence of the modular

function. To get i ts usual expression, i t suff ices to use the remark ( i i ) fol lowing

1.12.

Proof of the proposit ion : By remark a, we may assume that J£is a constant Hi lber t

bundle and that F(jC=) = L2(GO,~,K).

( i ) Let us check that L(f) is a wel l -def ined bounded operator. The map

x ~ f (x ) (L(x) ~od(x), nor(x)) is measurable and dominated in absolute value by

I f ( x ) l l~od(x)I Inor (x) l . This last function is vo- integrable, because

~0 = D-1/2v' - 1 = D - I and use of the Cauchy-Schwarz inequal i ty y ie lds

~ I f ( x ) l I~od(x)I Inor(x) l d~o(X)

_< [ I l f ( x ) l l~od(x)l 2 du-Z(x)] I /2 [ j I f ( x ) l I~or(x) l 2 dr(x) ] 1/2

< [ ! I f ( x ) I d~u(X) I~(u)I 2 du(u)] 1/2 [ ~ I f (x ) l d~U(x) In (u) l 2 du(u)] 1/2

1/2 I /2 -< Nfll I,d I~I IIfIll, r Inl

_< IIfIIi I~I I~I

Therefore, x w f(x) (L(x) ~od(x), nor(x)) is vo-integrable and L(f) is a bounded

operator of norm IL(f)II _< IIfII1- The continuity of the map L : Cc(G ) ÷~(r(J~)) follows

from the previous line. We have to check that L is a .-homomorphism. So, let f,g be

in Cc(G ). On one hand, (L(f . g)~,q) is equal to

~( i f ( x y ) g(y-1) o(xy,y-1)dxd(X)(y) (L(x) ~od(x), qor(x)) D1/2(x) dXu(X)d~(u)

54

= i f (xy) g(y-1) ~(xy,y-1) (L(x) god(x), nor(x)) D1/2(x) d~2(x,y).

The use of Fubin i 's theorem is j u s t i f i e d since we are in tegrat ing loca l l y integrable

funct ions on compact sets. On the other hand, we may wr i te the equation

L(g)g(u) : ~ g ( y ) k(y) god(y) D-I /2(y) d~U(y) ~ a.e. , so that (L( f )L(g)g,n)

is equal to ~ f ( x ) g ( y ) (L(x) (k(y) god(y), nor(x)) D-1/2(y)d~d(X)(y) dXu(X)du(u ).

Sett ing (x,y) ÷ (xy,y -1) and using a resu l t in the proof of 1.3.3, we obtain

f (xy)g(y -1) a(xy,y -1) (L(x) gor(y) , nor(x)) Dl /2(x) d~2(x,y). This shows that

L(f , g) = L( f )L(g) . Next, for f s Cc(G ),

( L ( f * ) g , n ) = ~ f (x -1) a(x,x -1) (L(x)

=

= . I f ( x ) (gor(x) , L(x)

= (g ,L( f )n ) .

~od(x), ~or(x)) d~o(X)

.f f (x ) ~ (x - l , x ) (L(x -1) gor(x) , nod(x)) duo(X),

(by symmetry of ~0).

~f(x) a ( x - l , x ) a(x,x -1) (gor(x) , L(x)nod(x)) d~o(X )

nod(x)) d~o(X)

F ina l l y , the representation L is non-degenerate. Indeed, le t n be a vector of F ( ~ )

such that (L( f )g ,n) = 0 for every f s Cc(G ) and every g c r ( J £ ) . Then, (L(x)

god(x), nor(x)) = 0 for ~ a . e . x . Choosing a countable to ta l set in K, one sees that

nor(x) = 0 for ~ a.e.x.Hence n(u) = 0 for ~ a . e . u .

( i i ) Let (u,Jg,L) and (~ ' , Jg ' , L ' ) be two equivalent a-representat ions. Let g

be a posi t ive loca l l y integrable funct ion on (GO,u) such that u' = g~ and ~ an

isomorphism of;}gontoJg' in ter tw in ing L and L ' . For g s r (jc~) define g' s r ,(jg')

by the formula g ' (u) = g -1/2(u) #(u) g(u). Then, the map g ÷ gl is an isometry, also

denoted ~, of r (J~) onto r , ( j £ ' ) which intertwines the integrated representations

L and L' . For,

L( f )g(u) = ~f (x) L(x) god(x) D- i /2(x) d~U(x)

L ' ( f ) g ' ( u ) = ~ f ( x ) L ' (x ) g'od(x) D ' - i / 2 (x ) d~U(x) and

D'(x) = gor(x) D(x) (god(x)) -1 v a.e. (1.3.3) . Thus (L ' ( f )~g) (u ) is equal to

! f ( x ) L ' (x) (god(x) - I / 2 ~od(x) god(x)(gor(x)) -1/2 D- I /2(x) (god(x)) I /2dxU(x)

: ~ f ( x ) g(u) - I / 2 ¢(u) L(x)

: ( @ L ( f ) ~ ) ( u ) .

55

~od(x) D-I/2(x) d~U(x)

Q.E.D.

I t is now easy to construct a f a i t h fu l family of bounded representations of

Cc(G,o ), namely the regular representations.As in the case of a group, they play an

essential role in the theory of groupoids. They have been defined and thoroughly

studied by P. Hahn in [45], where i t is pointed out that they have been long-time

favor i tes to produce von Neumann algebras (by the so-called group-measure space

construction).

Let ~ be a 2-cocycle and u a quasi- invar iant measure. Consider the measurable

f i e l d of H i lber t space {L2(G,~U), u E G O } with square integrable sections

I m L2(G,~ u) du(u) = L2(G,v). For x E G, define L(x) mapping L2(G,~ d(x)) to

L2(G,~r(x)) by L(x)~(y) = ~ ( x , x - l y ) ~ ( x - l y ) . This y ie lds a G-representation of G

(cf . example 3.11 of [45]) :

( L ( x ) L ( y ) C ) ( z ) : ~(x , x - l z ) ( L ( y )~ ) ( x - l z )

~ ( x , x - l z ) ~ ( y , y - l x ' I z ) ~ ( y - l x - l z )

~ ( x , y )o ( xy , y - l x - l z )~ ( ( xy ) - l z )

: o(x,y) L(xy)g(z).

The argument 1.1 shows that the function (L(x) god(x), nor(x)) =

. I ~ ( x , x - l y )~ (x - l y ) n(Y) d~r(X)(Y) is a continuous function of x for ~,n ~ Cc(G). Since

any vector in L2(G,~) is a pointwise l i m i t of a sequence in Cc(G ), th is function is

measurable when ~ and n are in L2(G,~).

1.8. Def in i t ion : The above c-representation of G w i l l be called the a-regular re-

presentation of G on ~. I ts integrated form is the regular representation on ~ of

Cc(G,o)-

I t is a basic fact ( [45] , theorem 2.15) that the regular representation on ~ is

the l e f t representation of a l e f t H i lber t algebra. We reproduce i t in our context.

The main ingredient of the proof, which is the construction of a l e f t approximate

56

i d e n t i t y , w i l l be needed in other places.

1.9. Propos i t ion : The algebra Cc(G,o ) has a l e f t approximate i d e n t i t y ( fo r the i n -

duct ive l i m i t topology) .

Proof : Let us say tha t a subset A of G is d - r e l a t i v e l y compact i f A n d - l (K) is

r e l a t i v e l y compact fo r any compact subset K of G 0. Then, i f L is r e l a t i v e l y compact,

AL = ( A n d - l ( r ( L ) ) ) L is also r e l a t i v e l y compact. Let us show tha t G O has a fundamental

system of d - r e l a t i v e l y compact neighborhoods. Let V be an open neighborhood of G O and

(Ki) a l o c a l l y f i n i t e r61a t i ve l y compact open cover of G O ( in G). There ex is ts a

r e l a t i v e l y compact open set U i in G such that K i c U i c V n d - l ( K i ) . Then U = u U i is

an open neighborhood of G O contained in V and is d - r e l a t i v e l y compact. Indeed, since

any compact subset K of G O meets only a f i n i t e number of K i ' s , U n d - l (K) is con ta i -

ned in a f i n i t e union of U i ' s . Let (Us) be such a fundamental system, w i th U s c U 1

fo r every s and l e t (Ks) be a net of compact subsets of G O increasing to G 0. We can

f i nd non-negative gs c Cc(G ), pos i t i ve on K s and wi th support contained in U s and

non-negative h ~ Cc(G° ) such that ha(u ) = ( Ig~d~U) - I f o r u E K s. Let us def ine

f (x) = h s or (x) g~(x). Then, f ~ Cc(G ), supp f c U~ and ~( fs ) = 1 on K s. We

claim tha t ( f s ) is a l e f t approximate i d e n t i t y . Let f ~ Cc(G ) w i th K = suppf. Then

s u p p ( f . f ) and suppf are contained in the compact set L = UIK. I f ~ > 0 is g iven,

the using the compactness of L and the c o n t i n u i t y of f ,~ and the product, one may

f ind s 0 such that fo r ~ > s 0 and every (x ,y ) ~ L × U nG 2, I f ( y - l x ) - f ( x ) I < ~ and

l { ( y , y - l x ) - I I < Ewh i l e r (L) c K s. I t fo l lows tha t

fs * f ( x ) - f ( x ) = l f ( y ) [ f ( y - l x ) - f ( x ) ] ~ ( y , y - Z x ) d ~ r ( X ) ( y )

+ f ( x ) • i f ( y )E~ (y , y - l x ) - I ] d~ r (X ) (y ) , and

I f m * f ( x ) - f ( x ) l ~ c+ sup I f (Y ) [ ~ fo r x c L. Y

Q.E.D.

I f ( f s ) is a l e f t approximate i d e n t i t y , ( f s * ) is a r i g h t approximate i d e n t i t y .

I have not been able to prove the existence of a two-sided approximate i d e n t i t y f o r

Cc(G ) except in p a r t i c u l a r cases ( r - d i s c re te groupoid and t ransformat ion groups).

The d e f i n i t i o n of a general ized H i l be r t algebra, used in the next p ropos i t i on ,

57

can be found in [73] , pages 5 and 6.

1.10. Proposit ion : (Cf. theorem 2.15 of [45 ] ) . Let ~ be a 2-cocycle and ~ a quasi-

invar iant measure. Then

( i ) Cc(G,~ ) with the inner product of L2(G,~ -1) is a generalized Hi lber t

algebra ; and

( i i ) i t s l e f t representation is equivalent to the regular representation on ~.

Proof : Let us check the axioms of [73].

( I ) For f ,g and h in Cc(G,~ ),

(g, f * * h ) = Ig(y ) f * * h(y) dv- l (y )

=J~g(y) f * (yx) h(x -1) o(yx,x -1) d~d(Y)(x) d~u(y ) d~(u)

=JJJg(y) f * (yx) h(x -1) o(yx,x - I ) d~r(x)(y ) d~U(x) du(u)

(meuse of Fubin i 's theorem is j u s t i f i e d because the funct ion (x,y) ~ f (y )

f * ( y x ) h(x-1)o(yx ,x -1) , defined on G u x Gu, is continuous with compact support)

=JJ/g(y) f * (yx - I ) h(x) o (yx - l , x ) d~d(x)(Y) d~u(X) d~(u)

= / / / f ( y -1 ) f * (y-Zx-1 ) h(x) o ( y - l x - l , x ) d~d(X)(Y) d~u(X) d~(u)

= / / fg(y-1) f ( x y ) o ( y - l x - l , x y ) h(x) o(y- lx - l ,x )d~d(X)(Y) d~u(X) d~{u)

: I I I f ( x y ) ~ y - 1 ) o(xy,y-1)d~d(X)(y) h(x) d~u(X ) d~(u)

: ( ~ . g, h). ( I I ) For every f s Cc(G ), g~ f . g is continuous. In fact , th is operator has

norm ~ I l f l l i , as i t can be seen d i rec t l y or deduced from ( i i ) .

( I I I ) Since Cc(~,~ ) has a l e f t approximate i den t i t y , the set { f * g : f , g ~ Cc(G)}

is dense in Cc(G ) with the induct ive l i m i t topology and a f o r t i o r i with the L2(G,u - I )

topology.

(IX) We have to show that the invo lu t ion , as a real l i near operator, is closable.

Suppose that fn 0 and f * 2d~-1 * n g. Then J I fn l 0 and f g(x)I 2 dv - l ( x ) = ÷ ÷ ÷ Ifn(X) - -1

~tfn(X) - g*(x) I 2 dr(x) ÷ O. Thus there is a subsequence fnk such that fnk÷ 0 v a.e.

and f ÷ g *va .e . Since ~ and - 1 are equivalent, g* = 0 va.e. , hence g = O. n k

( i i ) Let us cal l L' the l e f t representat ion on L2(G,~-I) , L ' ( f ) g = f , g, and

L the regular representation on u acting on L2(~,v). The isometry from L2(G, ~) onto

58

L2(G,v -1) which sends ~ into ~' = D 1/2 ~ implements the i r equivalence. For

~,n c L2(G,~) and f c Cc(G,v),

( k ' ( f )C ' , n ' ) = - ~ f , ~ ' (y) n ' ( y ) dv- l (y )

= ISJf(x) ~ ' ( x - l y ) ~(x,x-ZY) n ' (y ) d~r(Y)(x) d~u(y) d~(u)

= ~ f ( x ) D1/2(x- ly) ~(x-!y) ~ (x , x - l y ) D1/2(y) n(Y)

d~r(Y)(x) d~u(y ) d~(u)

=J~ f ( x ) ~ (x - l y ) { ( x , x - l y ) n(y) O-I /2(x) d~r(Y)(x)d~U(y) du(u)

= ~ f ( x ) ~ (x - ly ) { (x ,x -Zy) n(y) d~r(X)(y) D-1/2(x) d~U(x) d~(u)

= ~ f (x ) (L(x) ~od(x), nor(x)) duo(X )

= (L ( f )¢ ,n ) .

Q.E.D.

By looking at the polar decomposition of the invo lu t ion , we obtain the usual

ingredients of the Tomita-Takesaki theory : the modular invo lu t ion J : L2(G,v - I )

÷ L2(G,u -1) is given by J~(x) = Dl /2(x) C(x -1) ~ ( x , x - l ) , and the modular operator A

is defined on L2(G,v) n L2(G,u - I ) by A~(x) = D(x) ~(x).

1.11. Proposit ion : Cc(8,~ ) has a fa i t h fu l family of bounded representat ions, con-

s i s t ing of regular representat ions.

Proof : Let u be a quas i - invar iant measure with induced measure u and le t L be the

regular representation of Cc(G,o ) on p. The kernel of L is { f ~ Cc(G,o ) : f vanishes

on suppu}. For, i f f vanishes on suppu, the formula 1.7 (*) shows that L( f ) = 0 ;

while conversely, i f f * g = 0 in L2(G,u -1) for any g c Cc(G ), then using a r igh t

approximate i den t i t y , one sees that f = 0 in L2(G,~-I) , so that f vanishes on suppl.

To conclude, we observe that G O has a f a i t h fu l family of quas i - invar iant measures,

the t rans i t i ve measures (de f in i t i on 1.3.9).

Q.E.D.

1.12. Def in i t ion : Let ~ be a 2-cocycle. The C*-a lgebra C~(G,~) is the completion

of Cc(G,e ) for the C*-norm defined in 1.5. I t is cal led the o-C'algebra of the

groupoid G. The C*-a lgebra of G is C*(G) = C*(G,1) .

5g

Remarks :

( i ) I t results from 1.2 that cohomologous 2-cocycles give isomorphic C*-a lgebras.

( i i ) In the case of a group or a transformation group, our de f in i t ion does not

quite agree with the usual one (eg. [24] p. 35) because of the absence of a modular

function in the involut ion. However, the C e-algebras are isomorphic. To see th is ,

l e t G = U x S. We denote by Cc(G ) the . -a lgebra of 1.1 and by Cc, (G) the . -a lgebra

of [24]. The involut ion for the l a t t e r is f . (u,s) = f * ( u , s ) A(s-1), where ~ is

the modular funct ion of the group S. Then the map from C~(G) to Cc.(G ) sending f to

f ' ( u , s ) = f ( u , s ) A - I / 2 ( s ) is a ~-isomorphism. I t extends to an isomorphism of

C*(G) onto C. (G) .

( i i i ) I f G is second countable, then Cc(G ) with the inductive l im i t topology is

separable, therefore C ~(G,o) is separable.

I f h is a bounded continuous funct ion on G O and f c Cc(G ), we define

hf(x) = hor(x) f ( x ) , and

fh(x) = f (x) hod(x).

Then, hf and fh c Cc(G ) and the fol lowing relat ions hold in the w-algebra Cc(G,o).

For every f , g E Cc(G,o ),

f * hg = fh * g,

h(f . g ) = hf * g, and

(hf) * = f 'h 'where h*(u) = h(u).

For example,

f , hg(x) = Sf(xy) hg(y -1) ~(xy,y-1)d~d(X)(y)

~f(xy) hod(y) g(y- l ) ~(xy,y- l ) d~d(X)(y)

f fh (xy ) g(y- l ) ~(xy,y-Z) d~d(X)(y)

fh * g (x)

In other words, h acts on Cc(G,~ ) as a double cent ra l izer (cf . [47]) , Moreover i t acts

continuously with respect to the inductive l im i t topology.

1.13. Lemma : I f L is a representation of Cc(G,a ), there exists a unique representation

60

M of Cc(GO) such that fo r every h ~Cc(GO ) and every f e Cc(G,o ) , L(hf ) = M(h)L(f ) and

L( fh) = k ( f )M(h) .

Proof : Let H be the space of the representat ion L and H 0 the l i near span of n

{L ( f ) ~ : f ~ Cc(G,~), ~ E H}. Let us t ry to def ine M(h) on H 0 by M(h) (~ L ( f i ) ~ i ) n I

= ~ L (h f i )~ i , j u s t as in [47], page 317. To show that M(h) is well I

def ined, i t su f f i ces to prove

n n

Z L ( f i ) ~ i : 0 =>Z L(hf i )C i : O. 1 1

Let ( f ) be a l e f t approximate i d e n t i t y fo r Cc(G,o ). Then

n n

L(h f i )~ i = l im Z L ( h ( f ~ ) * f i ) ) ~ i : l im 1 1

n

= l im L(hfa) ~ L ( f i ) ~ i = 0 .

Moreover, M(h) satisfies

n

L(hf * f i )~i l

(M(h)L( f )~ ,L(g)n) : ( L ( h f ) ~ , L ( g ) n ) = (~ ,L (h f ) *L (g )n )

= ( ¢ , k ( f * * h*g)n) = ( ~ , k ( f * ) k ( h * g ) n )

= ( k ( f ) ~ , k ( h * g ) n ) = ( k ( f )~ , M(h*)k(g)n) .

To show tha t M(h) is a bounded operator , one uses as in [24] , page 41, the re la t i on

(hg) . (hf) +(kg) . (k f ) = !!hII 2 g . f

valid for every h ~ Cc(G0), f, g E Cc(G,~), where k(u) = (llhll 2 - lh(u)I2) I/2. Then

n

IIM(h) Z L ( f i ) ~ i N2 = Z (L (h f i )~ i , L ( h f j ) ~ j ) 1 i , j

*

= .Z. (L((hfj . (hfi))Ci,~j) 1,J

= , , k * * llhll2 1,0X ( L ( f j * * f i ) ~ i ' ~ J ) - i,j~ ( L ( f j ) ( k f i ) ) ~ i , E j )

=llhll 2 II~. k ( f i ) ~ i l l 2 - II ~k (k f i )~ i l l 2 I i

-< Ilhll 2 II !L(fi)Cill 2" 1

Therefore M(h) extends to a bounded operator on H. It is then routine to check that H

is a representat ion of the * -a lgebra Cc(GO ) and that L( fh) = L( f )M(h) . Q.E.D.

61

1.14. Proposition : The C*-algebra C*(G O ) is a subalgebra of the mu l t i p l i e r algebra

of C *(G,a).

Proof : The action of Cc(GO ) as double central izers of Cc(G,~ ) extends to C (G,o),

because for every bounded representation L of Cc(G,~ ),

IIL(hf)ll ~ NM(h)II l lL(f) l l ~ Ilhli Hfll , hence

Hhfl[ £ llhN ] l f l l .

This gives a*-homomorphism of C ~(G O) into the mu l t i p l i e r algebra of C*(G,~) which

is v i s i b l y one-to-one.

Q.E.D.

The notion of generalized expectation used in the next proposition was introduced

by M. Rief fel in [63] (de f in i t ion 4.12) in a context close to this one. We shall Iook

at i t again in the second section.

n

1.15. Proposition : The res t r i c t i on map Cc(G,a ) ÷ Cc(G~ ) is a generalized expectation.

I t i s smooth and f a i t h f u l .

The proof w i l l be given in a more general s i tuat ion in the second section (2.9).

Remark : I f G is r -d iscrete , C*(G O) is a subalgebra of C*(G,a) and the res t r i c t i on

map of Cc(G ) onto Cc(GO ) extends to an expectation of C*(G,~) onto C~(GO). In this

case, C*(G,a) has a uni t i f f G O is compact.

I t w i l l be convenient in the fol lowing discussion to enlarge the class of

functions on G.

1.16. Proposition : Let B(G) denote the set of bounded Borel functions on G with

compact support. With convolution and involut ion defined as in 1.1, B(G) can be made

into a* -a lgebra , denoted B(G,a).

The proof is s imi lar to 1.1. One can also use 1.1 and the fact that any function

in B(G) is a bounded pointwise l i m i t of a sequence of functions in Cc(G ).

Moreover, we may define the fol lowing notion of convergence in B(G,a) : a

62

sequence ( fn ) in B(G,~) converges to f c B(G,a) i f f fn(X) ÷ f ( x ) f o r every x ~ G

and there ex is ts h E B(G) such tha t I fn l ~ h and I f l ~ h. Then fn + f " gn ÷ g ~ > f n

gn ÷ f * g and f * ÷ f * n

Let us def ine a representat ion of B(G,~) as a *-homomorphism L : B(G,a) ÷ ~ ( H ) ,

÷ f -----> )~,~) ÷ where H is a H i l be r t space, which is continuous in the sense fn (L( fn

(L ( f )E ,n ) f o r any ~,n ~ H, and is such that the l ine-at.span of { L ( f )~ , f c B(G,~),

c H} is dense in H.

1.17. Lemma : Every representat ion of C (G,o) extends to a representat ion of B(G,o). C

Proof : Suppose that f ~ B(G). There ex is ts a sequence ( fn) in Cc(G ) converging to f

in B(G,o). By Lebesgue's dominated convergence theorem, fo r every ~,n in the space

H of the representat ion L, f is in tegrab le w i th respect to the measure ( L ( ) ~ , v ) and

(L ( fn )~ ,~ ) ÷ ( L ( f ) ~ , n ) . By the uni form boundedness theorem, L ( f ) is a bounded operator .

To show tha t L is a *-homomorphism, we use again an approximation argument. The

c o n t i n u i t y of L resu l t s from Lebesgue's dominated convergence theorem.

Q.E.D

The next goal is to r ea l i ze the inverse semi-group ~b of n@n-singular Borel

G-sets (1.3.26) as an inverse semi-group of pa r t i a l isometr ies. For S ~ b and

f ~ B(G), we def ine

s f ( x ) = ~ I / 2 ( r ( x ) , s ) f ( s - l x ) ~ ( s , s - Z x ) i f x ~ r - l ( r ( S ) ) ,

= 0 i f x # r - l ( r ( S ) ) ;

f s ( x ) = 6 1 / 2 ( d ( x ) , s - l ) f ( x s - 1 ) a ( x s - 1 , s ) i f x e d -L (d(S) ) ,

= 0 i f x ~ d - l ( d (S ) ) ; and

s* f = a ( s - l , s ) ( s - l f ) ,

where ~ ( . , s ) denotes the ve r t i ca l Radon-Nikodym de r i va t i ve of S. The notat ions have been

def ined in 1.1.11 and 1.1.18. For convenience, o ( s , s - l x ) is w r i t t e n instead of

o ( s r ( s - l x ) , s - l x ) . In accordance w i th 1.1.18,

~ ( s , t ) (u) = a ( u s , ( u . s ) t ) .

Also fo r a bounded Borel func t ion h on G O and f ~ B(G), h f ( x ) = hor(x) f ( x ) . We note

tha t s f , fs and s * f are funct ions in B(G).

63

1.18. Lemma : The following relations hold in the *-algebra B(G,a) :

( i ) s( t f ) = ~(s,t) (s t ) f for s, t C~b and f E B(G) ;

( i i ) fs~g = f.sg for f,g ~ B(G) and s ~ b ;

( i i i ) ( f s ) *= s* f ' f o r f ~ B(G) and s ~ b ;

( iv) s ( f~g) = s f . g for f,g ~ B(G) and s e~b ; and

(v) fn ÷ f ---~Sfn ÷ sf for fn' f ~ B(G) and s ~ b -

Proof : The verif ications are straightforward computations.

( i ) s( t f ) = ~I /2(r(x) ,s) t f (s - lx ) a(s,s- lx) for x ~ r - l ( r ( s t ) )

= ~Z/2(r(x),s) ~ i /2( r (x ) -s , t ) f ( t - l s - l x ) { ( t , t -Zs - l x ) { ( s ,s - l x )

= a l /2( r (x) ,s t ) f ( t - l s - l x ) ~(s,t)Qr(x) ~(s t , (s t ) - l (x )

= ~(s,t) (st) f (x), and

= 0 for x ~ r - l ( r ( s t ) ) .

( i i ) fs.g(x) = ~fs(y) g(y-lx) ~(y,y-lx) d~r(X)(y)

= ~ 61/2(d(y),s-Z)f(ys-l)g(y-Zx) ~(ys-l ,s) ~(y,y-Zx)d~r(X)(y).

Changing the variable y into ys, this last expression becomes

~I/2(d(y).s,s-1) ~(d(y),s) f(y) g(s-Zy-Zx) ~(y,s)~(ys,s- ly- lx)

d~r(X)(y)

= ~ 61/2(d(y),s) f(y) g(s- ly-Zx)~(y,y- lx)~(s,s- ly- lx) d~r(X)(y)

= ~ f(Y) sg(y-lx) o(y,y-Zx) d~r(X)(y)

= f . s g ( x ) .

( i i i ) (fs)*(x)= fs(x -1) ~-(x,x - I )

= a l /2 ( r (x ) ,s - l ) f~x-ls - I ) m-(x-ls-I s) ~--(x,x -1)

= 61/2(r(x),s - I ) f-(x-ls - I ) ~(x,x- ls - I ) # ( r (x )s - l , ( r (x ) .s -1)s)

= al /2(r(x),s-Z ) f (x-ls -1) ~(sx,x-ls-1)~(s-1 sx) ~(s-Z,s)or(x)

= 61/2(r(x),s -1) f (sx){(s- l ,sx) ~(s- l ,s )or (x)

= ~ ( s - l , s ) ( s - I f ) ( x ) .

(iv) s(f .g)(x) = ~l /2(r(x) ,s) f . g (s-lx)

=~61/2(r(x) ,s) f (s- lxy) g(y- l ) ~(xy,y-1) d~d(X)(y)

= ja l /2 ( r (xy ) ,s ) f (s- lxy) g(y-1) ~(xy,y-1) d~d(X)(y)

= s f , g (x ) .

64

(v) This is c lea r , since we assume tha t the v e r t i c a l Radon-Nikodym de r i va t i ve

a( . ,s ) is bounded on compact sets.

Q.E.D.

1.20. Lemma : Let L be a representat ion of B(G,~).

( i ) There is a unique representat ion M of B(G O) such tha t L(h f ) = M(h)L( f ) and

L( fh) = L( f )M(h) fo r every h ~ B(~ O) and every f ~ B(G).

( i i ) There is a unique G-representat ion V of the Borel ample semi-group ~b

as an inverse semi-group of pa r t i a l isometr ies such that L (s f ) = V(s )L ( f ) and

L( fs ) = L ( f )V(s ) fo r every s ~ ~b and every f ~ B(G).

( i i i ) The fo l l ow ing covariance r e l a t i o n between V and M holds : V(s) M(h) V(s ) *

= M(h s) f o r every s ~ ~b and every h e B(G O) where hS(u) = h(u s) i f u e r ( s ) ,

= 0 i f u # r ( s ) .

Proof : We f i r s t note tha t the approximate i d e n t i t y constructed in 1.9, which can be

chosen countable since G is second countable, s a t i s f i e s e n . f ÷ f fo r any f ~ Cc(G,~ ),

Let L be a representat ion of B(G,~) on the H i l b e r t space H and l e t H 0 be the l i n e a r

span of {L ( f )~ : f e Cc(G), ~ E H}. We proceed as in 1.13 t o ' de f i ne M(h) and V(s)

on H 0 : n n

M(h) ( ~ m( f i ) ~i ) = ~ L ( h f i ) ~ i , and 1 1 n n

= ~ L ( s f i ) ~ i • V(s) ( I k ( f i ) ~ i ) 1

We check as in 1.13 tha t M(h) and V(s) are wel l def ined.

( i ) This is obtained as 1.13.

( i i ) I t is immediate to check the f o l l ow ing re l a t i ons on H 0 :

V(s) V( t ) = M(~(s , t ) ) V(st )

V(s) - I = M( -~s , s - l ) ) V(s -1)

V(s) V(s) -1 = M(Xr(S) ) and V(s) - I V(s) = M(×d(S) ), where ×A is the c h a r a c t e r i s t i c

func t ion of A ; and

V(s) -1~ V(s)* In p a r t i c u l a r , V(s) is a pa r t i a l isometry and i t extends to H.

( i i i ) For s S~b, h E B(G O) and f ~ B(G)

65

s h-~-(s-l,s) s -1 f (x ) =61/2(r (x) ,s)

= ~ l /2 ( r ( x ) , s ) h ( r (x ) .s ) 7 (s - l , s )

= h ( r (x ) ,s ) f (x ) ~ (s - l , s ) a(s,s -1)

= (hSf) (x).

( h ~ ( s - l , s ) ) s - l f ( s - l x ) a ( s , s - l x )

i / 2 ( r ( x ) . s , s -1) f ( x ) ~ (s - l , x ) a (s ,s - l x )

Therefore,

V(s) M(h) V(s)* L(f) = V(s)r1(h) M(-~(s-ls))

= L ( sh -~ (s - l , s ) s - l f )

= L(hSf) = M(h s) k ( f ) .

V(s - I ) L( f )

Q.E.D.

We have seen(1.7) that bounded representations of Cc(G,a ) could be obtained by

integrat ing a-representations of G. The correspondence between the unitary represen-

tat ions of a group and the representations of i t s C *-algebra is well known and

j u s t i f i e s a large part of the theory of C*-algebras. The generalize~ion of this

resul t which we give in the case of groupoids has a more l imi ted scope. We only

consider groupoids which are second countable and representations on separable

Hi lber t spaces. Moreover, we need an addit ional assumption on the groupoid, namely,

i t should admit s u f f i c i e n t l y many non-singular G-sets (de f in i t ion (1.3.27). This

assumption is sa t is f ied in the case of a transformation group and of an r -d iscrete

groupoid. I do not have any example where i t is not sa t i s f ied . The case of a trans-

formation group is not new (e.g. [74], theorem 9.11, page 73). However, the proof

usually given uses the standard Borel structure of the group and seems to f a i l in the

case of a groupoid. Instead, we w i l l use part of a theorem of P.Hahn ( [43] , theorem 5.4,

page 106), which w i l l be reproduced below as part of the proof of 1.21 since i t has

not yet appeared in pr in t .

1.21. Theorem : Let G be a second countable loca l l y compact groupoid with Haar

system and with s u f f i c i e n t l y many non-singular Borel G-sets and a a continuous

2-cocycle in Z2(~,~). Then, every representation of Cc(G,¢ ) on a separable Hi lber t

space is the integrated form of a a-representation of Go

66

Proof : We w i l l only consider f ac to r representat ions. The general case is then ob ta i -

ned by d i r e c t in tegra l decomposition and requires the d e f i n i t i o n of a d i rec t in tegra l

of a fami ly o f ~-representat ions of G. Since th is theorem w i l l only be used in the

case of fac to r representat ions, we omit the general case here. Let L be a fac to r

representat ion of Cc(G,o ) on the separable H i l b e r t space H. We use 1.17 to extend i t

to a representat ion o f B(G,~) and 1.20 to obta in the representa t ion M o f B(~ O) and

the ~- representat ion V o f ~b such that L(hf) = M(h)L(f) and L(sf) = V(s )L ( f ) . I t

resu l ts from m u l t i p l i c i t y theory that there ex is ts a p r o b a b i l i t y measure ~ on GO and

a H i l b e r t bundle ~ o v e r (GO,u) such tha t M is u n i t a r i l y equ iva len t to m u l t i p l i c a t i o n

on the H i l b e r t space F(~£) of square- in tegrab le sect ions. From now on, ~e assume that

H = ~(~) and that M is m u l t i p l i c a t i o n .

a. Our f i r s t task is to show that the measure u is quas i - i nva r i an t . Let v be i t s

induced measure. We show tha t fo r f E B(G), i f f = 0 v a .e . , then L( f ) = O. Let E =

{u E G O : J l f l d ~ u > 0}. By assumption, M(XE) = O. I f x # r -1 (E) , then fo r

eve~I g ~ B(G),

f * g(x) = ~f(y) g(y-1) a ( y , y - l x )d~ r (X ) ( y ) = 0 and there fore

f , g = XE(f . g). Then

L ( f )L (g ) : L ( f , g) = M(×E) L ( f * g ) = O.

Since L is non degenerate, L( f ) = O. Thus, fo r f ~ B(G), L ( f ) depends only on the

v-c lass o f f . To show that u is quas i - i nva r i an t , we pick a Borel set A in G of pos i t i ve

v-measure and we show that i t has pos i t i ve v- i-measure. We may assume that fo r every .

u e r(A) and any open set V in G such that V n A u # ~, ~U(v m A) > N. Since G has

s u f f i c i e n t l y many non-s ingular Borel G-sets, there ex is ts a non-s ingular Borel

G-set S o f pos i t i ve u-measure which is contained in A. We can construct a sequence

(Un,en) where U n is a Borel set contained in A and e n a non-negative funct ion in B(G)

vanishing outs ide U n such that fen d~ u = I f o r u e r(A) and every n and (Un) shrinks

to S in the sense tha t every neighborhood of S contains U n f o r n s u f f i c i e n t l y large.

Let fn (y ) = ~ l / 2 ( r ( y ) , s ) en(Y ) fo r y E r - l ( r ( S ) ) , 0 otherwise. Then, fo r every

f ~ Cc(G ),

- I r(x) fn ~ f ( x ) =J~m/m(r (y ) ,s ) en(y ) f ( y - l x ) ~(y ,y x)d~ (y)

= J ~ l / 2 ( r ( x ) , s ) en(Y ) f ( y - l x ) ~ ( y , y - l x ) d ~ r ( X ) ( y ) .

67

Hence, fo r every x,

fn *' f ( x ) ÷ ~ i / 2 ( r ( x ) , s ) f ( s - l x ) a(s ,s-Zx) = s f ( x )

Moreover Jfn ~" f l ( x ) _< J s f I ( x ) . Therefore, L(fnO)L(f) = L ( f n . f ) ÷ L ( s f ) = V(s )L ( f )

in the weak operator topology. I f A had zero v - I - measure, then since supp f n c f ~ l ,

.¢-x- we would have 'n = 0 v a.e. and L ( f n ) * = L( fn) = O, hence L( fn ) = O. We would conclude

tha t V(s )L ( f ) = 0 fo r every f ¢ Cc(G ) , hence V(s) = O. However th i s would con t rad i c t

the fac t tha t V(s)V(s) = M(×r(S) ) > O.

b. Let us show next tha t fo r each non-s ingu lar Borel G-set S, the pa r t i a l isometry

V(s) on r (~) is of the form

V(s)~ (u) = z~l/2(us,s) c (u ,s) ~(u-s) f o r u ~ r(S)

= 0 fo r u f r(S)

where & ( ' , s ) is the hor i zon ta l Radon-Nikodym de r i va t i ve of S (1.3.18) and c (u , s ) ,

def ined fo r ~ a.e. u E r(S) is an isometry of ~Z~u. s onto~Zu u. This fo l lows d i r e c t l y

from a r e s u l t of Guichardet ( [38 ] , p ropos i t ion i , page 82) which we reca l l here :

L e t ~ a n d ~ b e two H i l b e r t bundles over the standard measure spaces (X,m) and

(Y,B) respec t i ve l y , ¢ an isomorphism of (X,m) onto (Y,6) and U an isometry of r(}~)

onto r (~) s a t i s f y i n g

UM(h) U -1 = M(ho¢ -1) f o r every h ~ L~(X,~),

where M denotes the m u l t i p l i c a t i o n operator. Then, there e x i s t isometr ies u(y) from

~Z~_l(y)Onto ~y def ined f o r B a . e , y such tha t f o r every ~ ~ r ( ~ ) ,

U~ (y) = r l / 2 ( y ) u(y) ~ ( ¢ - 1 ( y ) ) B a . e .

where r = dCm is the Radon-Nikodym de r i va t i ve of ¢~ wi th respect to B. dB

c. Next, we show tha t the set o f constant m u l t i p l i c i t y A = {u e G O : dim~Q = p}

f o r p = 1,2 . . . . . ~, of the H i l be r t b u n d l e ~ i s almost i n v a r i a n t ( d e f i n i t i o n 1 .3 .5) . I f

A were not almost i n v a r i a n t , there would be a Borel set B of pos i t i ve ~-measure such

tha t fo r every x E B, r ( x ) c A and d(x) # A. By assumption, B contains a non-s ingu lar

Borel G-set S such tha t ~ ( r (S) } > O. However f o r ~J a.e. u E r (S ) , there is an isometry

c (u ,s ) from ~Ju.s onto ~ u ' hence u-s E A. This is a con t rad i c t i on .

d. We show tha t fo r a Borel set B in G O , the p ro jec t ion M(XB) is in the commutant

68

of L i f f B is almost i nva r ian t . Suppose that there ex is ts a Borel subset A of G such

that M(B) L(XA) # L(×A) M(B). Then v(A n r - l ( B ) & A n d - l (B ) ) > 0 : B is not almost

i nva r i an t . Conversely, i f B is not almost i n v a r i a n t , then e i t he r r - 1 (B ) \d - l (B ) or

d ' 1 ( B ) \ r - l ( B ) has pos i t i ve v-measure and contains a non-s ingular Borel G-set S wi th

~( r (S)) > O. Then M(×B)V(s ) = V(s) and V(s)M(×B) = O. Since V(s) is the weak closure

o f { L ( f ) : f ~ Cc(G)} , there ex is ts f c Cc(G ) such that M(XB)L(f ) ~ L(f)N(×B). Since

we assume that L is a fac to r representa t ion, th is shows that ~ is ergodic. From c,

we conclude that the H i l be r t bundle~Y~is homogeneous, hence isomorphic to a constant

H i l be r t bundle, so that we can wr i te H = L2(GO,u,K).

e. We show that L sa t i s f i es the i nequa l i t y

l ( L ( f ) ~ , ~ ) I ~ j l f l d v 0 II~!l l!nll fo r every ~,n ~ K where ~ ~ K is i d e n t i f i e d wi th

the constant sect ion u ÷ ~. Let ~ and n be f i xed un i t vectors in K. Since by a. the

measure f ÷ (L ( f )~ ,n ) is abso lu te ly continuous wi th respect to VO, there ex is ts a

Borel func t ion c such that (L ( f )~ ,n ) = i f ( x ) c(x)dvo(x ) fo r every f ~ B(~). We have

to prove tha t Ic (x ) l < I v a.e. I f not, there e x i s t a > I such that v{x e G :

I c (x ) l > a} > 0 and we may f ind a Borel set A contained in {x c G : I c ( x ) l > a}o f

pos i t i ve v-measure and such that fo r u E r(A) and every open set V which meets p u,

~U(v n A) > O. Proceeding as in a, we f ind a non-s ingular Borel G-set S o f pos i t i ve

u-measure contained in A and a sequence (Un,en) where U n is a Borel set contained in

A and e n a non-negative funct ion in B(G) vanishing o f f U n such that

( i ) ~e dX u = I fo r u ~ r(A) " n

( i i ) U n shrinks to S when n ÷ ~ ; and

( i i i ) fo r every y c Un, D(s ' l y ) ~ b 2 where i < b < a.

I t is possible to f u l f i l l th is l as t cond i t ion because any neighborhood of a subset

o f G O of pos i t i ve ~-measure contains a subset of pos i t i ve v-measure where D ~ b 2.

Let fn(y ) = ~ I / 2 ( r ( y ) , s ) e n ( Y ) I ~ ( y ) / I c ( y ) l . Then

(L( fn)~ ,n) = J fn(y) c(y) D-m/m(y)dv(y)

= ~61/2( r (y ) ,s ) D-m/2(r(y)s) en (Y ) I c ( y ) ID -1 /2 ( s ' l y ) dr (y)

=#51 /2 (u ,s ) D-1/2(us) en (y ) I c (y ) I D -1/2 (s - l y ) d~U(y) du(u)

=~f& l /2 (u ,s ) en(Y) Ic (y ) I D - I / 2 ( s - l y ) d~U(y) d~(u)

69

by (1.3.20) and this dominates

with

ab -1 / A1/2(u,s) du(u).

r(S)

On the other hand, we know from b that

V(s)~(u) = A1/2(u,s) c(u,s) ~(u's) for u~ r(S)

c(u,s isometry of~{o, s into ~u" So

(V(s)~,~) = J ~ l /2(u ,s) (c(u ,s)~,n) d~(u) ; and

I (V(s)~ 'n) I # ~r(S) &l /2(u 's) du(u).

This is a contradict ion because L(fn) : M(hn) V(s), where

hn(u ) =fen(Y) E(}]/ Ic(y)I d~U(y) sat is f ies ]hnl ~ ~ I ,

tends to zero in the weak operator topology. Indeed,

f~ . f (x ) - h~sf(x) =/e ( y ) i ~ i ~ l / 2 ( r ( x ) , s ) [ o ( y , y - l x ) f ( y - l x ) -

~ ( s , s - l x ) f ( s ' I x ~ dxr(X)(y)

tends to zero for every x e G and every f e Cc(G), and

I f a * f ( x ) l ~ l ( s f ) ( x ) I and lhmsf(x)l ~ ] ( s f ) ( x ) I .

f . The conclusion is given by the fol lowing lemma, due to P.Hahn ( ~ 9 ] , theorem

5.4, page 106).

Lemma (P. Hahn) : Let G and ~ be as above. Let L be a representation of Cc(G,~) on

L2(GO,~,K) where u is a quasi- invar iant probabi l i ty measure and K a separable Hilberi

space, such that

( i ) l (L ( f )~ ,n) I ~ J I f l d v 0 II~II Ilqll for every E,n~K

( ~ also denotes the constant function ~L2(GO,u,K) ) .

( i i ) L(hf) = M(h)L(f) for every h ~ Cc(G O) and every f e Cc(G ), where M is

mul t ip l ica t ion.

Then, L is the integrated form of a ~-representation of G on the constant Hi lber t

bundle with f iber K over (GO,u).

Proof :

a. There exists a weakly measurable map x ÷ A(x) of G into the bounded operators

70

on K of norm < i such that

(L( f )~ ,~) = J f ( x ) (A(x)~,n) du0(x ) for every ~,n c K.

For the condit ion ( i ) means that f ~ (L( f )~ ,n) is a bounded l inear functional on

LI(G,~o) of norm ~ II~II llnII. This gives a map (~,n) ~ k(~,n) of K × m into L~°(G,~).

This map is sesqui l inear and sa t i s f i es Ik (~,n) l~ ~ ll~II IInll. Using a l i f t i n g of L~(G,v)

into the space ~ ( G , v ) of bounded measurable funct ions, we obtain for each x c G

a bounded sesqui l inear functional on K × K of norm ~ 1, hence an operator A(x) of norm

< 1. The map x ~ A(x) has the required propert ies.

b. For every ~,n ~ L2(~O,~,K) and for every f E Cc(G ),

(L( f )~ ,n) = Y f ( x ) (A(x)~ od(x ) , nor(x)) duo(X).

Since both sides are bounded sesqui l inear funct ionals on L2(G 0 ,~, K) (cf . 1.7) i t

suf f ices to check the equal i ty on the algebraic tensor product L2(G0,~) ® K and by

sesqu i l inear i ty on elements of the form h(u)~, where h E Cc(G0 ) and ~ ~ K :

(L( f )h~,kn) = (L(f)M(h)~, M(k)n)

= (k(k* fh)~,n) by ( i i )

:jk- or(x) f ( x ) hod(x) (A(x)~,~) dvo(X )

=J f (x ) (A(x) hod(x)~, kor(x)n) du0(x )

c. A sa t i s f i es A ( x ) * = # (x ,x -1) A(x -1) for u a . e . x . For a l l ( ,n e K and f s Cc(G )

(k ( f * )~ ,n ) = J f * ( x ) (A(x)~,n) dvo(X )

= ~ f ( x - I ) T(x ,x -1) (A(x-Z)(~,n) dvo(X )

= ~ f ( x ) @(x ,x -1) (A(x ' l ) (E ,n) dvo(X ) (because VO is symmetric), and

(~,L(f)m) = f f ( x ) (~, A(x)n) dvo(X )

Hence ( A ( x ) * ~ , n ) = ~(x,x -1) (A(x)~,n) for u a.e.x.

the resu l t .

d. The function A sa t i s f i es A(x)A(y) = ~(x,y) A(xy) u 2 aoe. (x ,y ) .

For a l l ~,n ~ L2(GO,~,K) and f ,g ~ Cc(G )

L(g)~(u) = j g(y) A(y) ~od(y) D- I /2(y) d~U(y) for

(k ( f )L(g)~,n) = J f ( x )g ( y ) (A(x)A(y)~od(y) ,nor(x) )

(These computations have already been done in 1.7.) On the other hand,

Since K is separable, we obtain

a.e. u by b and

D ' I /2 (xy ) D- l (y) du2(xy).

71

(L ( f * g)g,q) = ~ f ( x y ) g ( y ' 1 ) o ( x y , y - I ) (A(x)~ od (x ) , nor (x) ) D1/2(x) dv2(x ,y) .

A f te r the change of var iab le (x,y) ÷ (xy ,y -1 ) , th is is equal to

f ( x ) g(y) ~(xy,y -1) (A(xy)~od(y) , qor(x) ) D1/2(xy) D- I (y) d~2(x,Y)

Using the densi ty of Cc(G ) ® Cc(G ) in Cc(G2), we obtain fo r a l l ~,q e K

(A(x)A(y)~,q) = ~(x ,y) (A(xy)~,q) fo r v 2 a.e. (x ,y)

since K is separable, we obtain our resu l t .

e. Let S be a non-singular Borel G-set. Since A cannot be evaluated on S, we consider

instead a funct ion B def ined as fo l lows. I t resu l ts from d that A(xy)A(y -1) =

a(xy,y " I ) A(x) for v 2 a.e. (x ,y) or equ iva len t l y , for v a.e. x and ~d(x) a . e . y .

Let B(x) = f A(xy) A ( y - l ) ~ ( x y , y -1) f ( d ( x ) , y ) d~d(X)(y) , where f is a pos i t i ve measu-

rable funct ion on G O x G such that

J f ( u , y ) d~U(y) = 1 fo r every u E G O .

Then, B(x) = A(x) fo r v a . e . x . Let us show that B(xs-1)B(s) = a ( x s - l , s ) B(x)

fo r v a .e .x . (As usual s in B(s) and in o ( x s - l , s ) stands fo r d ( x s - l ) s ) . By quasi-

invar iance o f u under S, i t resu l ts from d that fo r v a.e. x c d - l ( d ( s ) )

A(xs- l )A(y ) ~ ( x s - l , y ) A ( x s - l y ) f o r x d (xs - l ) = a.e. y and

B(xs -1) = A(xs-1). Therefore

B(xs-1)B(s) = A(xs -1) J A(sy)A(Y - I ) E (sy ,y -1 ) f (d (x ) ,y )d~d(X) (Y)

= A(xs " I ) f A(y)A(Y - I s ) ~ ( y , y -Zs ) f ( d ( x ) , s -Zy ) d~d(x) 's -Z(y)

= ] A ( x s ' l y ) A ( y - l s ) o ( x s - l , y ) ~ - ( y , y - l s ) f ( d ( x ) , s - l y ) d ~ d ( x ) ' s - l ( y )

=SA(xy )A (y -1 )o (xs - I , sy ) ~ (sy , y - I ) f ( d ( x ) ' y ) d~d(X)(Y)

= o (xs 'Z ,s )B(x ) .

f . A(x) is a un i tary operator fo r u a . e . x . Hahn's proof uses the von Neumann

se lec t ion lemma. We w i l l use instead the existence of s u f f i c i e n t l y many non-singular

G-sets. The set E = {x E G : B(x) is not un i ta ry } is a measurable set. Suppose that

i t has pos i t i ve v-measure. Then, i t contains a non-singular Borel G-set S of pos i t i ve

-measure. Let us def ine v l ( s ) on L2(GO,~,K) by

( , ) v l ( s ) (u) = AZ/2(u,s)B(us)~(u 's) i f u ~ r(S) and

= 0 otherwise.

72

Then for every f s Cc(G ),

L ( f )v l (s )~(u) = J f (x )B(x ) A l /2(d(x) ,s ) B(dx)s) ~(d(x).s)D-1/2(x ) d~U(x)

for ~ a,e. u.

We change the variable x into xs -1 to obtain

~f(xs-1)B(xs - I ) A l /2 (d (x ) .s - l , s )B(s )~(d (x ) ) D-1/2(xs - I ) 6(d(x),s-1)d~U(x).

We use A(u,s-1)D(us - I ) = a(u,s - I ) for u a.e. u (1.3.20) to obtain

5 f (xs -1) a l /2(d(x) ,s -Z) B(xs -1) B(s) ~(d(x)) D- i /2(x) d~U(x).

Final ly by d, this y ields

J f(xs - I ) 61/2(d(x),s -1) a (xs- l ,s ) B(x) ~od(x) D-I /2(x) d~U(x)

= I f s ( x ) A(x) ~od(x) D-I /2(x) dxU(x)

= L(fs)~(u).

Hence L ( f ) v l ( s ) = L(fs) = L(f)V(s) for every f , so that VI(s) = V(s). In par t icu lar

v l (s) is a non-zero part ia l isometry with range M(×r(S) ). Comparing (*) and b of the

proof of the theorem, we see that B(us) is a unitary operator for ~ a.e. u in r(S).

This is a contradict ion. Hence, for ~ a.e. x, B(x) and consequlntly A(x) are uni tary.

I t suff ices to modify A on a null set so that i t becomes unitary for every x to

f u l f i l l a l l the conditions of de f in i t i on 1.6.

Q.E.D.

1.22. Corollary : Under the assumptions of the theorem, every representation of Cc(G,a )

on a separable Hi lber t space is bounded.

1.23. Corollary : Under the assumption of the theorem, the integrat ing process 1.7

establishes a b i jec t i ve correspondence between (a,G)- H i lber t bundles and separable

Hermitian C*(G,a)-modules which preserves inter twin ing operators.

Proof : We have to prove that two a-representations (~,J£~L) and (u ' , J~ ,L ' ) which give

un i ta r i l y equivalent integrated representations are equivalent. Let L and L' be repre-

sentations of Cc(G,a ) on r(J~) and F ( ~ ) whose res t r i c t ions to Cc(G O) are mul t ip l ica-

tions M and M'. I f # is an isometry of F(~C) onto r ( ~ ) which intertwines L and L ' ,

i t also intertwines M and M'. Therefore, the scalar spectral measures u and u' of M

and M' are equivalent and there exists a measurable f i e ld u ~ #(u) where #(u) is an

73

isometry of ]~u o n t o ~ , decomposing ~. The relat ion {L( f ) = L ' ( f )~ becomes

~f(x)(~ or(x)L(x)~od(x) , nor(x))d~o(X ) =~ f (x ) (L ' ( x )~d (x )~od(x ) ,no r (x ) ) d~o(X ) where

we have assumed ~ = ~'. This gives ¢or(x) L(x) = L ' (x ) {od(x) for v a . e . x .

Q.E.D.

The o-representations of a group G can be considered as ordinary representations

of the extension group G °. This leads to an alternate def in i t ion of C*(G,o), for a

groupoid G. Let G a denote the extension]~x G of -[by G defined by the 2-cocycle

o ~ Z2(G,q[). Recall (1.1.12)that

(s,x) ( t , y ) = (s to(x ,y) ,xy)

( s , x ) - I = ( s - l o ( x , x -1 ) - l , x -1)

I ts uni t space can be ident i f ied with G O . I t is a local ly compact grou:ooid with the

product topology and i t has the l e f t Haar system {h x ~u} where h is the Haar

measure of 7[.

1.22. Proposition : The C*-algebra C*(G,o) is the quotient of C*(G °) by the kernel

I of the representation L of C*(G °) obtained by integration which sat is fy

L( t f ) = tL( f ) for any t sT, f ~ Cc(G° ) and where t f ( s ,x ) = f ( t ' l s , x ) .

Proof : The map ~ from Cc(G °) to Cc(G,o ) given by the formula ~f(x) = ] f ( s , x ) sds

is a *-homomorphism. Indeed,

~( f*g)(x) = f f *g (s ,x )sds -i -1 =J]J f ( s to (x ,y ) , xy )g ( t "o(y,y )-Z,y-l)dtd~d(X~(y)sds.

One makes the changes of variable u = sto(x,y) and v = t - l a ( y , y - l ) - I to obtain

jJf f (u,xy) g(v,y -1) uvo(y,y-1)o(x,y)-Zdudvdxd(X)(y)

= J ( j f ( u , x y ) u d u ) ( j g(v,y-1)vdv)o(xy,y -1) d~d(X)(y)

= ~(f) . ~(g) (x).

Moreover,

~ ( f * ) ( x ) = # f * ( s , x ) s d s

= ] - f ( s -1a(x ,x - l ) -1,x-1)sds

=~ f ( t , x -1) t dt ~(x,x -1)

= ~( f ) (x - I ) o(x,x -1)

74

= ~T(f) (X) ,

Since ~ is bounded with respect to the L I norms, i . e .

l l~( f ) I I l ~ ! I f I l l , because

f l~ ( f ) (x ) [d~U(x) ~JJ [ f ( x , s ) I dsd~U(x), i t fo l lows that i t is bounded

with respect to the C *-norms and extends to a *-homomorphism from C*(G ° ) to

C * ( 8 , ~ ) . I t is onto since i t s image is closed and contains Cc(G,o ). I f L is a repre-

sentat ion of C ~ (G,~), Lo~ is a representat ion of C* (G ~) which s a t i s f i e s

Lo~ ( t f ) = k ( t ~ ( f ) ) = t Lo~(f) since ~ ( t f ) = t ~ ( f ) .

Conversely, i f L is a representat ion of C* (G°), sa t i s f y i ng th is re la t i on and which

is of the form

(*) (L( f )E,q) = J J f ( s , x ) (L(s ,x)~od(x) , nor(x)) dv0(x)ds ,

then L(s ,x) s a t i s f i e s

L ( t s , x ) = tL (s ,x )

fo r h ×h a.e. ( t , s ) and ~ a.e. x, as one sees from the equation L(gf) = ~(g)L( f )

where g e Cc(T) ,

( g f ) ( s , x ) = ~ g ( t ) f ( t - l s , x ) d t and ~(g) = ~ g ( t ) t d t .

L(s ,x) can be replaced by

L ' ( s , x ) : J L ( t s , x ) t d t

wi thout changing ( , ) . Then i t s a t i s f i e s

L ( t s , x ) = tL (s ,x ) fo r every ( t , s ) and v a . e . x . Thus

(L ( f )~ ,n ) = j j f ( s ,x )sds (L (e ,x )~od(x) ,nor (x ) ) d~0(x),

so that L factors through ~.

Q.E.D,

2. Induced Representations

Let G be a l o c a l l y compact groupoid with Haar system {~u} and H a closed sub-

groupoid G containing G O and admit t ing a Haar system { ~ } , and ~ ~ Z2(G,T) a continuous

2-cocycle. Just as in the case of groups, a u-representat ion of H may be induced to a

75

o - rep resen ta t i on of G, We descr ibe th i s process below, R i e f f e l ' s vers ion [54] of indu-

ced representa t ions is p a r t i c u l a r l y wel l adapted to th is contex t and th is sec t ion

im i ta tes the expos i t i on he gives in the case of groups. We f i r s t need some topo log ica l

r esu l t s which are wel l known in the group s i t u a t i o n .

2.1. Propos i t ion : Let G and H be as above and consider the r e l a t i o n on G def ined by

x ~ y i f f d(x) = d(y) and xy -1 ~ H.

( i ) I t is an equivalence r e l a t i o n .

( i i ) The quo t i en t topology on the quo t i en t space H\G is Hausdorf f .

( i i i ) The quo t i en t map r : G ÷ H'G is open.

( i v ) The quo t i en t space H\G is l o c a l l y compact.

(v) The domain map d induces a continuous and open map from H\G onto G O .

Proof :

( i ) Clear .

( i i ) Since H is c losed, the set { ( x , y ) ~ G 2 : xy ~ H} is closed in G 2, hence in

G x G. The graph o f the r e l a t i o n is the image by e o f t h i s se t , where e is the

homeomorphism ( x , y ) , + (x ,y - I ) : G × G ÷ G x G

( i i i ) Let 0 be an open set G ; we have to show tha t i t s sa tu ra t i on HO is a lso open.

Let hx be a po in t in HO wi th h ~ H and x c O. There ex is ts a noncnegative func t ion

¢ E Cc(H ) such tha t ¢(h) ~ 0 and a non-negat ive func t ion g E Cc(G ) such tha t

g(x) # 0 and supog c O. The same argument as in 1.1 shows tha t the func t ion ¢-g

def ined by

¢'g (Y) = I ¢ ( k ) g ( k - l y ) dX~ (y) (k)

is continuous on G ; t he re fo re {y : ¢ .g(y ) # O} is an open set ; since i t contains hx

and is contained in HO, we are done.

( i v ) This resu l t s from ( i i ) and ( i i i ) .

(v) This is c lea r s ince d : G + G O is continuous and open. Q.E.D.

2.2. Lemma : There ex i s t s a Bruhat approximate c ross-sec t ion fo r G over H\G, tha t

i s , a non-negat ive continuous func t ion b on G whose support has compact i n t e r s e c t i o n

w i th the sa tu ra t i on HK of any compact subset K of G and is such tha t f o r every x e G,

I b ( h - l x ) dz~(X)(h) = I . M

76

Proof : By [ ~ , Lemme 1, page 96, there ex is ts a non-negative continuous func t ion g

non-zero on every equivalence class and whose support has compact i n te r sec t i on w i th

the sa tu ra t ion of any compact subset o f G. The func t ion gO defined by g°(x) =

g (h - l x ) d ~ (x) (h) is continuous and s t r i c t l y pos i t i ve . The func t ion b = g/gO

is a Bruhat approximate cross sect ion fo r G over H\G.

Q.E.D.

2.3. Propos i t ion : Let G and H be as above and consider the r e l a t i o n ~ on G 2

def ined by ( x , y ) ~ ( x ' , y ' ) i f f y = y ' and xx ' - I E H.

( i ) I t is an open Hausdorff equivalence r e l a t i o n and the quot ien t space H\G 2

w i th quot ient topology is l o c a l l y compact.

( i i ) The re l a t i on is compatible wi th the groupoid s t ruc tu re of G 2, so tha t

HSG 2 is a l o c a l l y compact groupoid, i t s u n i t space may be i d e n t i f i e d w i th HIG.

( i i i ) The groupoid H\G 2 has a Haar system, namely {6~ × ~d(~), ~ ~ H\G}.

Proof :

( i ) This is v e r i f i e d as 2.1, in fac t H\G 2 = { (~ , y ) E H\G x G : d(~) = r ( y ) } .

( i i ) The composable pairs in G 2 are ( ( x , y ) , ( x y , z ) ) . Therefore i f ( x ' , y ) ~ ( x , y ) ,

then ( x ' y , z ) ~ (xy ,z ) and ( x ' , y ) ( x ' y , z ) = ( x ' , y z ) ~ (x ,yz) = ( x , y ) ( x y , z ) . Hence we

may def ine the fo l l ow ing groupoid s t ruc tu re on H\G 2. The composable pai rs are ( x , y ) ,

( ~ , z ) , (~,y) (x-y,z) = (½,yz) and the inverse of ( x ,y ) is (x~1 ,y -1 ) .F ina l l y , by

d e f i n i t i o n of the quot ien t topology, the m u l t i p l i c a t i o n and inverse maps are cont inuous.

Since (~,y) (# ,y ) -1 = (# ,yy-1) = ( x , d (~ ) ) , we may i d e n t i f y the u n i t s~.ace of H\G 2

and H\ G.

( i i i ) This is c lear ; here J f ~ ×d~ d(#) = I f ( ~ , y ) d x d ( X ) ( y )

Q.E.D.

Notat ion : Let o be a continuous 2-cocycle in Z2(G,-~), one can associate w i th i t

continuous 2-cocycles on H and on H\G 2 respec t i ve l y in the fo l l ow ing way. On H,

denotes i t s r e s t r i c t i o n to H. On H\G 2, ~ is def ined by ~ ( x , y , z ) = ~(y ,z) (we wr i t e

( x , y , z ) instead of ( ( x , y ) , ( ~ , z ) ) ) . The cocycle property is eas i l y checked.

For# ~ Cc(H,a ) and f c Cc(G), l e t us def ine

77

• f (x) = J ¢(h) f ( h - l x ) ~(h,h-lx) dx~(X)(h) , and

f • ¢ (x) = # f(xh) ¢(h -I) ~(xh,h -I) dz~(X)(h .

For ~ m Cc(HIG2,o ) and f c Cc(G), let us define

• f (x) = ] ¢ ( x - l , x y ) f ( y - l ) o ( x y , y - l ) d x d ( X ) ( y ) , and

f • ~ (x) = I f (Y) ~ (Y ,y - l x ) ~ (Y ,y - I x ) d~r(X)(y) .

2.4. Proposi t ion :

( i ) The space Cc(G ) is a Cd(H,~)-bimodule and a Cc(H~G2,o)-bimodule ; and the

act ions of Cc(H,{ ) and Cc(H~,G2,~) on opposite side commute.

( i i ) The algebra Cc(H,~ ) acts as a *-a lgebra of double cen t ra l i ze rs on the

algebra Cc(G,~), th i s act ion extends to the C* -a lgebra C*(G,~) and gives a

*-homomorphism of C* (H,~) in to the m u l t i p l i e r algebra of C (G,~).

Proof :

( i ) One has f i r s t to check tha t , wi th above nota t ions, Cf, f¢, m-f and f .# are

indeed in Cc(G ). This is done in exact ly the same fashion as in propos i t ion 1.1. The

v e r i f i c a t i o n of the var ious a s s o c i a t i v i t y r e l a t i ons ,

(¢ * 9) - f = ¢ - ( ~ . f ) for f c Cc(G )

and ¢,# both in Cc(H,{ ) or in Cc(H\G2,~), the analogous re la t i on for the act ion on the

r i gh t , and

• ( f • ~) = (¢ • f ) • ~ fo r f ~ Cc(G )

and ¢,~ in C (H,~) or in C (H\G2,~), is s t ra igh t fo rward but tedious. Let us check

one of them as an example. Suppose f ~ Cc(G ) and ¢,~ c Cc(HXG2,~). Then

• ~ (x ,y) = f ¢ (x ,yz )~(xyz ,z -1) o(yz,z-m)dxd(Y)(z) , and

f . ( ¢ . ~ ) (x) = J f ( y )¢ ~ (~ , y - l x ) ~ ( y , y - l x ) d l r ( X ) ( y )

= # f ( y ) ¢(!},y-mxz) ~(~z,z -1) ~ ( y , y - l x ) ~(y-mxz,z-m)

• d~d(x) (z)d~r(x) (y)

= # f ( y ) # ( y , y - l x z ) ~(x~,z - I ) ~(y,y-mxz) ~(xz,z -1)

d~r(x) (y) d~d(X)(z)

= I f • ¢(xz) ~ ( ~ , z - I ) ~(xz,z -1) d~d(X)(z)

=J f • ¢(z) # ( z , z - l x ) d ( z , z - l x ) d x r ( X ) ( z )

78

= ( f • #) • ~ (x) .

( i f ) We have to check the equations

f ~ ( ¢ . g ) = f • ¢ * g and (# . f)~' = f * . #~

for f , g e Cc(G ) and ¢ E Cc(H ). This is done as above. To prove that th is act ion extends

extends to C * ( G , { ) , one can introduce the Banach algebra L i , r ( G , a ) , the completion

of Cc(G,a ) fo r the norm ll l [ i , r . I t has a bounded l e f t approximate i d e n t i t y . Thus, i f

L is a bounded representat ion of Cc(G,a ), there is a unique bounded representat ion

L H, ca l led the r e s t r i c t i o n of L to C*(H,~), such that L(¢- f ) = LH(#)L(f ) and

L ( f .# ) = L ( f ) LH(#). What makes the proof go is the inequa l i t y I I# . f l I i , r ~ I I# j I i , r I I f I l l , r

which is obtained as in 1.6. This gives a f a i t h f u l ~-homomorphism of Cc(H,a) into the

m u l t i p l i e r algebra of C*(G,a) which is norm-decreasing when Cc(H,c ) has thell III norm.

Hence i t extends to a *-homomorphism of C*(H,~) in to the m u l t i p l i e r algebra of

C * (G,~).

Q.E.D.

Let X = Cc(G), B = Cc(H,~ ) and E = Cc(H\G2,~) ; view X as a l e f t E- and r i gh t

B-bimodule. One would l i ke to exh ib i t X as an E-B i m p r i m i t i v i t y bimodule ( d e f i n i t i o n

6.10 of [ 6 4 ) . I did not succeed in doing that except in pa r t i cu la r cases. The \

candidates fo r E and B-valued inner products on X are

<f ~ g>B (h) = j ~(x -1) g(x-Zh)~(x,x -1) ~ ( x , x - l h ) d x r ( h ) ( x ) and

< f , g > E ( X , x - l y ) = ~ f ( x - l h ) g(y ,h) ~(y - lh ) ~ ( y ' l h , h - l y ) a (x - l h ,h - l y )dx~ (X ) (h ) .

(By l e f t invariance of the Haar system, the r i gh t hand side depends on x on ly) . The

algebraic re la t ions

<f , gb> B = <f ~g>Bb ( f , g c X, b ~ B)

<ef,g> E = e<f ~ g>E (e E E)

<f,gb> E = <fb* 'g>E

<ef'g>B = < f 'e~ g>B

f<g ' f '>B = <f'g>E f ' ( f ' g' f ' c X)

are s a t i s f i e d , as one may check in the same fashion as above.

2.5. Lemma :

( i ) The l i nea r span of the range of <'>E contains an approximate l e f t i d e n t i t y

79

for Cc(HIG2, ~) with the induct ive l i m i t topology.

( i i ) A s im i l a r statement holds for <'>B and Cc(H,e ).

Proof : I t is the same proof as in proposi t ion 7.11 of ~3] (and in lemma 2, page

201, of ~9 ] ) .

( i ) Let C be a compact subset of H\G and c a pos i t i ve number.Choose a compact

set K in G such that ~(K) = C. There ex is ts a d - r e l a t i v e l y compact (see 1.9) neigh-

borhood N of G O in G such that l o ( x - l , y ) - 11 < ~ fo r y ~ ~, x ~ K and r ( y ) = r ( x ) ,

because ~ is continuous and takes the value I whenever one of i t s arguments is a un i t .

There is a l o c a l l y f i n i t e cover of G consist ing of open r e l a t i v e l y compact sets (Vi)

such that v i l v i c N and a p a r t i t i o n of un i ty subordinate to i t . Mu l t i p l y th is

pa r t i t i on pointwise with a Bruhat approximate cross-sect ion b which has been

truncated so that b ~ Cc(G ) and f b ( h - l x ) d ~ ( X ) ( h ) = 1 for x c K, and obtain a f i n i t e

number of non-negative funct ions of f l . . . . . fn c Cc(G) such that suppf i ~ V i and n

f f i ( h ' l x ) d ~ ( X ) ( h ) = 1 fo r x c K. For each i , choose gi ~ Cc(G) such that i= l

suppg i ~ V i , ~ Ig i (Y) Id~U(y) = 1 fo r u ~ r {x : f i ( x ) ~ 0} , and [ g i ( y ) I =

~ i ( y ) ~ ( y - l y ) . Let e fc c NI = ~ <~ ,g i> , where f ( x ) = f ( x ' l ) . Since ~ ~ ~ ' ' J i=1

< f i ' g i >(~ 'y) = ] f i (h - Ix ) gT (h-mxy) ~ ( y - l x - l h ' h -mxy ) { ( x - l h ' h - l x y ) d ~ ( x ) ( h ) ' e(c,e,N)

s a t i s f i e s

(a) 0 x ~ 0 i f y # N, and (c,~,N)( 'YJ =

(b) l ]O(c,~,N)(~,y)d~d(X)(y) -11 L ~ i f x cC.

I t resu l ts from the proof of proposi t ion 1.9 that the net {e (c ,~ ,N) )d i rec ted by

~ , ~ , N ) < ( c ' , ~ ' , N ' ) i f f c ~ c ' , c >c' and N ~N' is a l e f t approximate i d e n t i t y fo r

Cc(H\G2,o).

( i i ) This is done in a s im i l a r manner. Let K be a compact subset of G O , e a

pos i t i ve number and N an r - r e l a t i v e l y compact neighborhood of G O in G, One can f ind

an r - r e l a t i v e l y compact neighborhood U of G O and non-negative continuous funct ions f

and g on 6 such that UU - 1 ~ N, the support of g is compact and contained in U, whi le

g(x)d~U(x) = 1 for u ~ K, the support of f is contained in U and has compact

in te rsec t ion with the saturat ion HL of any compact subset L of G, whi le

I f ( h - l x ) d l~(X)(h) = 1 fo r x ~ r - l (m) n U, and I# (h -mx ,x - l h )o (h - l x , x -1) - 1! ~

when x ~ U n r - l ( K ) , h - l x ~ U and h ~ H. Then, one notes that the funct ion @(c,~,N)'

80

def ined by <~(N,E,K)(h) = <f,g>B(h)

= I f ( h x ) g(x) { ( h x , x - l h -1) a (hx , x -1 )d~d (h ) ( x ) ,

s a t i s f i e s

(a) ¢(N,E,K)(h) = 0 i f h ~ N and

(b) [ /¢(N,c,K)(h)dXHu(h ) - 1] E c i f u E K.

Therefore, the net {~(n,E,K)} is a r i g h t approximate i d e n t i t y fo r Cc(H,d ).

Q.E.D.

2.6. Coro l la ry :

( i ) The l i n e a r span of the range of <'>E is dense in Cc(H\G2,~) and in C* (H\G2,a)

C ~ ( i i ) The l i nea r span of the range of <'>B is dense in Cc(H,o ) and in (H,o).

Remark : I t seems d i f f i c u l t to const ruct approximate i d e n t i t i e s as those obtained

in lemma 2, page 201, of [39]. In the general case, wi th the notat ions of the proof ,

one would need sets V.'sl such that r (V i ) = G O . In the case when H = G 0, i t is not

hard to carry the proof through. This is done in the next p ropos i t ion .

2.7. Propos i t ion : Let H = G O , B = Cc(GO ) and E = Cc(G2,a). Then X = Cc(G ) is an

E-B i m p r i m i t i v i t y bimodule. In other words, C ~ (G2,~) and C* (G O ) are s t rong ly Mori ta

equ iva len t .

Proof : I t is not su rp r i s ing tha t th i s r esu l t is independent of { since G 2, being

cont inuous ly s im i l a r to G 0, has t r i v i a l cohomology. Thus we may assume tha t ~ 1. We

have to check the l as t con t i t i ons in the d e f i n i t i o n of an i m p r i m i t i v i t y bimodule. The

B-valued inner-product is c l e a r l y pos i t i ve :

<f , f>B(U) = J1 f ( y -1 ) I2d~U(y ) .

The E-valued inner-product is pos i t i ve ; as mentioned before, we can f i nd here an

approximate i d e n t i t y f o r the r i g h t act ion of Cc(G ) of the form <fK,fK>B ; namely,

l e t K be a compact subset of G O , g ~ Cc(G ) nonzero on K and h eCc(G O) such that

h(u) = I l l g(y-1)12d~'U]- l /2 f o r u E K ; then set fK = gh. To complete the proof we

need only v e r i f y the norm condi t ions

<fb'fb>E ~ IIbll2 <f ' f>E and <ef,ef> B ~ Nell 2 < f ' f>B

where e E E, b e B and f ~ X. But

81

<fb, fb>E(x,y ) = Ibor(x) I 2 <f , f>E(x,y) and

Hbll 2 <f,f>E - <fb,fb>E = <fc,fc>m

where c(x) = (IImll 2 - Ibor (x )12) l /2 . Assume that e c E is non-negative. Then

<ef,ef>B(U ) = J l e f (Y -m) Imd~u(y)

= I l l e ( y , y - l z ) l / 2 f ( z - 1 ) e ( y , y - l z ) l / 2 d x r ( Y ) ( z ) 1 2 d x U ( Y )

I r _< y / e ( y , y - l z ) I f (z - I ) 12d~r(Y)(z) Y'e(y,y ~z)d~, (Y)(z) d~,U(y)

< sup le(y,z)d~d(Y)(z)# e ( y , y - l z ) I f ( z - I ) 12dX r (z) (y) dxU(z) Y

<_ ~le~Ii,r (sup ye(zy,y-1)d~d(Z)(y)) J If(z - I ) i2d>,U(z) Z

_< !lelli, r IIeEII,d <f,f>B (u)

< ,e l I <f, f>B(U).

This gives a .-homomorphism L : E ~ L(X) where L(X) is the algebra of bounded

operators on the pre B-Hi lber t space X which is bounded when E has the IT Ill-norm ;

therefore, i t is normidecreasing. By d e f i n i t i o n of the C * -norm on E, IIL(e)II < llei!

and hence the required inequa l i t y .

OE.D.

A representat ion of C* (G O ) can be induced up to a representat ion of C*(G2,~)

by R i e f f e l ' s tensor product construct ion (Coro l lary 6.15 of [ 6 ~ ) and " res t r i c t ed "

* * G2,~ to C (G,o), which acts on C ( ) as double cen t ra l i ze rs (a funct ion on G can be

viewed as a funct ion on G 2 depending on the second var iab le only) . A l t e r n a t i v e l y , the

r e s t r i c t i o n map P : Cc(G,m ) ÷ Cc(GO ) is a general ized condi t ional expectat ion ( [63 ] ,

d e f i n i t i o n 4.12) and so a representat ion of C* (G O ) may be induced to C * (G,o) via P.

Let us construct e x p l i c i t l y these representat ions induced from the un i t space ( th is

w i l l be used in 3.2). For s i m p l i c i t y , consider a m u l t i p l i c i t y - f r e e representat ion of

* (G 0) L2(G 0 C , given by m u l t i p l i c a t i o n on the space ,~), where p is a measure on G O .

The space of the induced representat ion is obtained by completing Cc(G ) ®Cc~O)Cc(Q O) =

Cc(G ) with respect to the inner product

<f . h, g , k>: I P(g* * f)h : I u)

= ~f ® h){g ® k]dv -I

82

where, as usual, v =jdxUd;~(u). The induced representation, denoted by !ndp acts on

L2(G,v -1) by convolution on the l e f t : for f E Cc(G ) and ~,q s L2(G,v -1)

( Ind~(f)~,~) = f f (xy)~(y-m)~(x)~(xy ,y " I ) d~U(y)dXu(X)d~(u).

We have met this representation before, in the case when ~ is a quasi- invar iant

measure (proposit ion 1.10) : i t is the regular representation on ~z.

I t resul ts from proposit ion 1.11 that the function defined by NfiIre d =

sup IIL(f)II, where L ranges over al l representations induced from the uni t space, is

C * a C -norm on Cc(G,~ ) dominated by the -norm Nfil.

2.8. Def in i t ion : The reduced C * -algebra Cred(G,~ ) of G is the completion of

Cc(G,o ) for the reduced norm [I fire d.

I t is a quotient of C*(G,~) since the ident i ty map on Cc(G,~ ) extends to a

• -homomorphism of C (G,~) onto Cred(G,c~ ).

Representations induced from more general subgroupoids w i l l only be considered

in the context where theorem 1.21 applies. The notion of generalized conditional

expectation used the fol lowing proposit ion was introduced by M.Rieffel in [6~

(de f in i t ion 4.12). This is the piece of structure which allows the construction of

induced representations.

2.9. Proposition : Assume that G is second contable and that G and H have suf f i c ien-

t l y many non-singular Borel G-sets. Then the res t r i c t i on map from the pre-C*-algebra

Cc(G,~ ) to the pre-C -algebra Cc(H,o ) is a generalized condit ional expectation.

The fol lowing lemma shows the pos i t i v i t y of P ; i t is due to Blat tner in the

case of groups ( [ ~ , theorem 1, page 424). We fol low here [ 6~ , theorem 4.4.

2.10. Lemma : Let (u,~,L) be a 1-representation of H ; then, for any f ,g e Cc(G,1 )

and any ~,~ ~ I~(~) (the space of square-integrable sect ions), we have

(Lop(g** f )~,n) =~b(x ) (~ ( f ,~ ) ( x ) ,~ (g ,~ ) ( x ) )dv (x ) where b is a Bruhat approximate

cross-section for G over H\G, v =5~u d~(u) and

~( f ,~ ) (x ) = f (x ' l k )L (k )~°d(k ) AH I /2(k) d ~ (z) (k)" (&H denotes the modular function

of u re la t i ve to ( H , ~ ) . )

83

Proof : We have

(Lop(g* . f)g,q) J g * = * f(h) (L(h) god(h), nor(h)) dVHo(h)

where vHO is the symmetric measure of ~ relat ive to (H, x~),. This, in turn, equals

/ / # ( x - l h -1) f(x -1) (m(h) god(h), nor(h)) dxd(h)(x) dvmo(h)

=J/(Jb(k-lx) d~ (x) (k)) g(x-lh -1 f(x -1) (L(h) god(h), nor(h)) dxd(h)(x) dvmo(h)

=Jf~b(k-lx) ~(x-lh -1) f(x -1) (L(h) ~od(h),nor(h)) dxr(k)(x) d~(h)(k) dvmo(h )

The use of Fubini's theorem is legit imate, because the support of the function

(k ,x )÷b(k - lx ) g(x-lh -1) f(x - I ) is compact :

k -1 x E H suppf~n suppb. We make the change of variable x ~ kx in the last integral

to obtain

Jffb(x) g(×-lk-lh-1) f(×-lk-1) (L(h)~od(h), qor(h))d~d(h)(×)dx~(h)(k)dVho(h). .1/2 (h)d~Hud~(u) and change the order of integration ; this is !{e write dVHo(h ) = f~H

jus t i f ied as above. We get

J b(x) ~(x- lk- lh -1) f (x - lk - I ) (L(h)Cod(h),qor(h)) A~/2(h) d~Hr(x)(h)

dxd(k)(x) dX~(k) dp(u) .

We make the change of variable h ~ hk -1, .yielding

I b(x) ~(x-lh -1) f (x - l k -1) (L(hk -1) ~or(k),qor(h)) A~/2 (hk -1)

dXad(k)(h) dxd(k)(x) dvH(k ).

We use the fact that AHI(k) dvH(k ) = dvHl(k) to produce

b(x) g(x-lh -1) f (x - l k -1) (L(k-1)¢or(k),L(h-1)nor(h)) A~/2(hk)

d~Hu(h ) dxU(x) d~Hu(k ) d~(u).

Final ly, we change the order of integration and arrive at .1/2 lh-1 ) fb (x ) (~( f (x - lk -1)L(k -1)~° r (k ) ~H (k), g(x-

L(h-l)nor(h) A~/2(h)) dXHu(h ) d~Hu(k))dxU(x ) dp(u) .

The vector-valued function k ~ f (x - l k - I ) L(k-1)~or(k) &~/2(k) is ~Hr(x) integrable

because i t is measurable and i ts norm is integrable since l~or(k)[ A~/2(k) is local ly

integrable. Hence

~(f,~)(x) =~f(x-mk -1) m(k-1)~or(k) 5~/2(k) d~Hr(x)(k ) makes sense and

( r ( f ,~ ) (x ) , v(g,n)(X)) is equal to .1/2~ (f(x-lk-1)m(k-m)gor(k) A~/2(k), g(x-mh-1)L(h-1)nor(h) A H kh)) dIHr(x)(h)dXHr(x)(k).

Q.E.D.

84

Proof of the propos i t ion .

( i ) P is se l f ad jo i n t , because P ( f * ) = P ( f ) * fo r f c Cc(G,o ) by simple

ca l cu la t i on .

( i i ) P is pos i t i ve , i . e . , P ( f * * f ) ! 0 in C (H,o) fo r f s Cc(G,o ). To see t h i s ,

we f i r s t consider the case g = I . The lemma t e l l s us tha t L ( P ( f * * f ) ) ~ 0 fo r any

representat ion L of C (G,1) obtained by i n teg ra t i on .S ince C*(G) has a f a i t h f u l fami ly

o f such representat ions by theorem 1.21, P ( f * * f ) Z O. To deal wi th the case of an

a r b i t r a r y cocycle o, we consider instead G ° and H ° . Since H a is a closed subgroupoid

of G ° and G ° and H ° s a t i s f y the hypothesis of the p ropos i t ion , the r e s t r i c t i o n map Q

from Cc(G~,I) onto Cc(H°, I ) is pos i t i ve . Since the diagram

Cc(Ge,I) Q ~ C c H~,I) w i th ~ f (x ) = Sf (s ,x)sds

I ~mg(h ) = ]g (s ,h )sds

~H

Cc(G,~ ) P • Cc(H,~ )

commutes and since ~ is onto whi le ~H is cont inuous, P is also pos i t i ve .

( i i i ) P s a t i s f i e s the expectat ion property ; i . e . , P(¢- f ) = # * P ( f ) fo r f s Cc(G,o)

and # s Cc(H,o ), as can be seen immediately from the de f i n i t on of ¢ - f .

( i v ) P is r e l a t i v e l y bounded ; tha t i s , fo r every g s Cc(G), the map

f ~ P(g* , f , g) is bounded wi th respect to the C*-norms) . To see t h i s , proceed as

in propos i t ion 4.10 of [63]. F i r s t , one establ ishes the i n e q u a l i t y

P(f* * * g * g * f) ~ llgll 2 P(f** f) .

I t su f f i ces to show the i n e q u a l i t y when one evaluates both sides against a pos i t i ve

type measure, tha t i s , a measure v on Cc(H ) s a t i s f y i n g v ( f ~ f ) ~ 0 fo r any f s Cc(H).

Via the GNS cons t ruc t ion , a pos i t i ve type measure on G def ines a representat ion of

Cc(G,o), hence of Cc(G,o ) by theorem 1.20 ; in p a r t i c u l a r , i t is continuous wi th

respect to the C - topol ~,, Therefore, v°P is a pos i t i ve type measure on Cc(G,o) O ~ j .

The corresponding representat ion is given by convolu t ion on the l e f t ;

'~ ~ fo r g , f s Cc(G), IIg * fll~op ~ Ilgll m I]fll op

where ( , )voP is the inner-product defined by voP :

vop((g . f) . (g .f)) ~ llgll 2 voP (f** f) .

Then, one appl ies the general ized Cauchy-Schwarz i n e q u a l i t y of p ropos i t ion 2.9 of [63]

85

to conclude.

(v) Let (ek) be an approximate l e f t i d e n t i t y fo r Cc(G,o ) wi th the induct ive l i m i t

topology. Then, fo r any f e Cc(G,~ ), ( f - e k , f ) * . ( f - e k . f ) tends to 0 in Cc(G,~ )

and i t s r e s t r i c t i o n to H tends to 0 in Cc(H,~ ), hence in the C*-norm.

( v i ) The range o f P is Cc(H,~ )-

( v i i ) P is c l e a r l y f a i t h f u l ; f * * f = 0 on H ~ > f * * f = 0 on G O ~ > f = O.

Q.E.D.

This p ropos i t ion al lows a pa r t i a l answer to a question that has been avoided

un t i l now. Given a l o c a l l y compact groupoid, we have assumed the existence o f a Haar

system and kept i t f i xed . Most not ions int roduced, such as quas i - invar iance or the

convolut ion product, depend e x p l i c i t l y on the choice of such a Haar system. What is

the ro le of th is choice and can we f ind not ions independent of i t ?

2.11. Coro l la ry : Let G be a second countable l o c a l l y compact groupoid, ( ~ ) i = 1,2

two Haar systems wi th respect to which G has s u f f i c i e n t l y many non-s ingular Borel

G-sets and l e t ~ be a continuous 2-cocycle. Then, the corresponding C* -a lgebras

C (Gl,a) and C (G2,~) are s t rong ly Mori ta equ iva lent .

Proof : We set G = G 1 and view G 2 as the subgroup H. Then H\G 2 = G 1. We can use

propos i t ion 2.9 to show tha t X = Cc(G ) is indeed an E:B i m p r i m i t i v i t y bimodule wi th

E = Cc(H\G2,~ ) and B = Cc(H,~ ) as before. Propositon 2.9 gives the p o s i t i v i t y of

<f ' f>B as wel l as the norm cond i t ion <ef ,ef> B ~llell I <f ' f>B fo r e , f E Cc(G ). By

symmetry, s i m i l a r statements hold fo r E.

Q.E.D.

2.12. Example : Let X be a second countable l o c a l l y compact space. We have def ined

( i . 3 .28 . c ) the t r a n s i t i v e groupoid on the space X as G = X x X, wi th the groupoid

s t ruc ture given in 1.1.2 ( i i ) and the product topology. We know that a Haar system

on G is def ined by a measure ~ o f support X. I f X is uncountable and m is non-atomic,

then G has s u f f i c i e n t l y many non-s ingular Borel G-sets. Let us f i x m. Since the class

of m is the only i nva r i an t measure class and any representa t ion of G is a mu l t i p l e

86

of the one-dimensional t r i v i a l representat ion, the corresponding C*-a lgebra is iso-

morphic to the algebra of compact operators on a separable H i lbe r t space. Thus two

measures ~1 and ~2 give isomorphic C*-algebras but there is no canonical way to

construct an isomorphism.

I t would be in teres t ing to have an example where two Haar systems give non-isomor-

phic C*-a lgebras.

3. Amenable Groupoids

The notion of amenabil i ty for groups (see [41] or [30]) takes many forms and a

large part of the theory consists in showing the i r equivalence. Our goal is much more

l imi ted here. We shal l f i r s t consider measure groupoids and choose a d e f i n i t i o n of

amenabil i ty best suited to our needs. We seek a condi t ion ensuring that every repre-

sentation is weakly contained in the regular representation. Then the von Neumann alge-

bra associated to any representation is i n jec t i ve ; here, the proof is essen t ia l l y the

same as in [83], where R.Zimmer studied ergodic actions of countable discrete groups.

A notion of amenabil i ty is then given for l oca l l y compact groupoids with Haar system,

whose main advantage is that i t is eas i ly checked.Some examples are studied. Throughcut

th is sect ion, G designates a l oca l l y compact groupoid wi th a f ixed Haar system {~u}.

3.1. De f in i t i on : A quas i - invar ian t p robab i l i t y measure ~ on G O w i l l be cal led

amenable (we also say that (G,u) is amenable) i f there exists a net ( f i ) in Co(G )

such that

( i ) the functions u ~ S l f i l2d~ u converge to 1 in the weak * - topo logy of

L~(GO,~) and

( i i ) the functions x ~ f f i ( x Y ) ~ i (Y)dxd(X)(y) converge to 1 in the weak .

-topology of L~(G,~) where v is the induced measure of p.

This d e f i n i t i o n reduces to one of the equivalent de f i n i t i ons of amenabil i ty in

the case of a group ; namely, that the funct ion i is the l i m i t , uniformly on compact

sets, of funct ions of the form f . f% where f ~ Cc(G)(one has to use theorem

13.5.2 of [19]).

87

The t r a n s i t i v e measures on a pr inc ipa l groupoid provide examples of amenable

measures. Let us ind ica te b r i e f l y how the net ( f i ) can be constructed. Fix an o r b i t

~u One can choose an increasino net (Ki) ~u] and l e t u be the t r a n s i t i v e measure d . .

o f compact sets in ~ ] (with the topology given by the b i j e c t i o n d : G u ÷ ~ ] ) such

that uK i = [u]. Define f i by

f i ( x ) = ~(Ki ) -1/2 i f ( r , d ) ( x ) s K i x Ki,

= 0 otherwise. Then

f . * f~ (x ) = 1 i f ( r , d ) ( x ) s K i x K i 1

= 0 otherwise.

The funct ion f i is not in Cc(G ) but i t is in L2(9,v) (where u has been normalized)

and i t can be approximated in L2(G,v) by elements of Cc(G ).

3.2. Proposi t ion : Let u be a quas i - i nva r ian t amenable p r o b a l i l i t y measure on G O and

o a 2-cocycle in Z2(G,~). Then the in tegrated form of any a- representa t ion of G

l i v i n g on u is weakly contained in the regular representat ion on u of C*(G,a).

Proof : We fo l low Takai ( [70 ] , page 29). Let (U ,~u ,L ) be a a-representat ion of G.

A vector s ta te of the in tegra ted representat ion is of the form

~(f) = f f ( y ) (L (y )~od(y ) ,~or (y ) ) d~o(y ) fo r f s Cc(G )

where ~ is a un i t vector in F(~Q. Let ( f i ) be as in 3.1 and def ine @i(f) by

@i(f) = S ( S f i ( x ) - ~ i ( y - l x ) d ~ r ( Y ) ( x ) ) f ( y ) (k (y )~od(y) ,~or (y ) )dvo(Y) .

By 3.1 ( i i ) , @i(f) tends to @(f). Moreover, a rout ine computation al lows us to wr i te

the equation

~ i ( f ) = ~ f ( x y ) ( ~ i ( y - 1 ) , ~ i ( x ) ) a(xy ,y -1) d~U(y) d~u(X ) d~(u) ,

where ~ i (x ) is def ined by

~ i (x ) = D l /2 (x ) ~ ( x , x -1 )T i ( x ) k (x -Z)~or (x ) .

We recognize the expression fo r @i(f) as ( I ndM( f )~ i ,C i ) , where IndM is the represen-

ta t ion induced by the r e s t r i c t i o n M of L to C*(G O) (see end of 2.7). I t acts on the

space r(I~) of square- in tegrable sect ions of the H i l be r t bundle~(~ u = L2(G,~u) ® ~ on

(GO,u). Let us compute the norm of Ci"

II~ i l l 2 = ~I I~i (x) I I 2 dv - l ( x )

= ; [ f i ( x ) I 2 [ k o r ( x ) I I 2 du(x) (because D - dv ) d~-I

88

= ] l l~(u)l l 2 ( I f i ( x ) l 2 dxU(x)) dp (u ) .

By 3.1 ( i ) , II~ill tends to 1. From the inequal i ty l ~ i ( f ) I L lllndM(f)N II(i l I ll~iIl, we

obtain l { ( f ) l ~ lllndM(f)ll. Since IndM is a d i rec t integral of representations equi-

valent to the regular representation on # of C (G,~), i t is weakly contained in i t

and so is L.

Q.E.D.

3.3. Remark : One expects a converse ; namely, i f the integrated form of the t r i v i a l

one-dimensional representation of G l i v ing on u is weakly contained in the regular

representation on ~ of C*(G) , then ~ is amenable. Let us say that a continuous func-

t ion # on G is of posi t ive type (with respect to u) i f ~ ( f ) = I f ( x ) ~(x) dvo(X ),

f E Cc(G), defines a posi t ive l inear functional ~ on C~(G). For example, the function

1, which is associated with the vector state ~ ( f ) =~f(x)dvo(X ) of the one-dimensional

t r i v i a l representation (~,~u u = ~, L = I) is of posi t ive type. Let us determine the

posi t ive type functions associated to the vector states of the regular representation

( ~ ' ~ u = L2(G'~U)' L(x) )where L(x)~(y) = ~(x - ly ) . The vector ~ ~ Cc(G ) c L2(G,v)

gives the posi t ive type function

(k(x) ~od(x),~or(x)) = ~ ( x - Z y ) ~(y)d~r(X)(y)

= ~ . ~ * ( x - 1 ) .

Hence, i f our hypothesis holds, the state ~1 is a weak l im i t of states associated with

posi t ive type functions which are f i n i t e sums of functions of the form ~ , ~ * ( x - 1 ) ,

with ~ C c ( G ). I t is not hard to show that these posi t ive type functions can in fact

be chosen to be of the form ~ . ~ * ( x - 1 ) . Indeed one observes that for ~, f ,g c Cc(G ),

w~(f * g*) =J~(x) f . g* (x) dvo(X )

=J~(x) f . g* (x -1) dvo(X ) (where ~(x) = ~(x-1))

= (L(~)f ,g)

Hence, i f ~ is of posi t ive type, L(~) is a posi t ive operator. Then, using Kaplansky's

density theorem to approximate i ts square root, one obtains a net ( f i ) in Cc(G ) such

that L( f i . f~) ÷ L ( ~ ) i n the weak operator topology and L ( f i . f ~ ) ~ L(~). This implies

f * (x -1) converge weakly to that the posi t ive l inear functionals associated to f i * i

89

m@. To conclude, one would need to exh ib i t ~1 as a weak l i m i t of states associated

f * (x -1) wi th f i which are uni- with pos i t i ve type funct ions of the form f i * i e Cc(G)

formly bounded in L~(G,v). So far I have been unable to do th is .

3,4. Lemma : Let ~ be a quas i - invar ian t p robab i l i t y measure on G 0, Then u is

amenable i f f there ex is ts an approximate invar ian t mean on L~(G,v), that i s , a net

(g i ) of non-negative funct ions in Cc(G ) such that

( i ) the funct ions u ~ jg id~ u converge to 1 in the weak . - topo logy of L~(GO,~) ;

and

( i i ) the funct ion x ~ i g i ( x Y ) - g i (Y) Id~d(X)(y) converge to 0 in the weak

* - topo logy of L=(G,v).

Proof : The proof is essen t ia l l y the same as in the case of a group (e.g. [41], page

61). Let us s t a r t wi th ( f i ) as in 3.1 and def ine gi = I f i 12" The f i r s t property is

immediate. Using the inequa l i t y llal 2 - !b12~< ( la l + I b l ) ( l a - b l ) and Cauchy-Schwarz,

one obtains

J Ig i (xY) - g i (Y) Id~d(X)(y) < [S ( I f i ( xy ) I + I f~ (y ) I )2d~d(X) (y ) ] 1/2

[ f l f i (xy ) - f i (y ) Imd~d(x) (y ) ]1 /2 .

Let us set h i (u ) = I I f i ( y ) I 2 d ~ U ( y ) . The f i r s t member of the product is majorized by

2 I /2 [hi or(x ) + h i o d ( x ) ] i / 2

whi le the second is majorized by

[11 " h i ° r ( x ) l + 11 - h i ° d ( x ) I + 11 - f i * f l (x)I

+ I1 - f i * fT ( x -1 ) l ] 1/2

The s-topology on L~(G,~) is def ined by the semi-norms ~@(f) = ( f@tf[2dv) 1/2 where

~p is a non-negative element of LI(G,~). The f i r s t term goes to 2 in the s-topology

and is bounded in the L~(G,.~) norm and the second goes to O. Thus t h e i r product goes

to 0 in the s- topology and a f o r t i o r i in the weak * - topo logy . Conversely, s ta r t i ng

wi th (g i ) , we def ine f i = gi 1/2. Again, the f i r s t property of ~. R i is immediately

s a t i s f i e d . Using the inequa l i t y la - bl 2 _ < la 2 - b21, one obtains wi thout much t rou-

ble the est imate

I1 - f f i ( xY ) f - ( y )d~d(X) (y ) l _<

1/2 [J lg i (xY ) - g i ( Y ) I d ~ d ( X ) ( y ) + I i - Sg i (Y)d~r (X) (y ) I

+ 11 _ i g i (Y )d~d(X) (y ) l ] . Q.E.D.

90

3.5. Propos i t ion : Let ~ be a q u a s i - i n v a r i a n t amenable p r o b a b i l i t y measure on G O and

o a 2-cocycle in Z2(G,~). Any a- representa t ion of G l i v i n g on ~ generates an i n j ec -

t i v e yon Neumann algebra.

Proof : As mentioned e a r l i e r , the idea of the proof is in Zimmer [83] . The not ion of

amenabi l i t y we use - i t is more s t r i ngen t than Zimmer's - makes the proof eas ier . Let

( ~ , ~ u , L ) be a a- representa t ion of G ; L also denotes the in tegrated representat ion

on F(}6) given by ( k ( f ) ~ , n ) = ~ f ( x ) ( L ( x )~od (x ) , no r ( x ) )d~o (X ) ~,n c r(~6) f E Cc(G) ;

JK~ denotes the von Neumann generated by {L ( f ) • f s Cc(G)}; J~6' is i t s commutant and

~) is the algebra of decomposable operators on F(~z~). An operator A E ~ acts on F ( ~ )

by A~(u) = A(u)$(u) where A(u) is an operator on ~ . We note tha t ~ =

{A ~ ) : Aor(x) L(x) = L(x) Aod(x) fo r v a.e. x} . Tomiyama has shown that avon

Neumann algebra is i n j e c t i v e i f f i t s commutant is i n j e c t i v e ; in p a r t i c u l a r ~), which

is the commutant of a commutative von Neumann algebra, is i n j e c t i v e . We w i l l const ruct

a cond i t iona l expectat ion of 5Donto d~6' ; t h i s w i l l show tha t ~6', hence J~6, is

i n j e c t i v e . Let (g i ) be a net as in 3.4 and l e t M be a bound fo r sup S g i d~u" We u

def ine a l i n e a r map Pi : ~ ) ~ ) by

PiB(u) = S g i ( x ) k(x) Bod(x) k(x) 'Zd~U(x)

i . e . (PiB~,n) = f g i ( x ) ( L ( x ) B o d ( x ) k ( x ) - l $ o r ( x ) , n o r ( x ) ) d v ( x )

f o r ~,n ~ r (~ ) .

There is no problem checking tha t Pi is wel l def ined. Horeover since

IIPiB(u)II < IIBII ~ g i ( x ) d ~ U ( x ) , we see tha t

IIPiBIl < MIIBII.

We also note tha t Pi is pos i t i ve . The P i ' s are un i fo rmly bounded in norm. Hence there

is a subset converging to a bounded pos i t i ve l i n e a r map P in the f o l l ow ing sense. For

every pa i r of vectors (~,n) in F ( ~ ) and fo r every B in ~), (PiB~,n) tends to (PB~,n).

The r e s t r i c t i o n of P to JK~' is the i d e n t i t y ( i n p a r t i c u l a r , P is u n i t a l ) . For i f

A E ~ , then

(PiA~,n) = f g i ( x ) (Aor(x) ~ o r ( x ) , n o r ( x ) ) d v ( x )

= J ( A ( u ) ~ ( u ) , n ( u ) ) ( I g i ( x ) d x U ( x ) ) d u ( u )

By 3.4. ( i ) , we obta in at the l i m i t ,

91

(PAd,n) = j (A(u )~(u ) ,n (u ) )d~(u ) = (A~,n).

The proof that P is an expectat ion w i l l be completed when we show that P(~)) = ~ .

A f te r rout ine computations, one obta ins, fo r B e ~ fo r f ~ C~(G) and ~,n ~ r (~ ) ,

(L( f )P iB~,n) =

~ f (x )g i (y )~(x ,y )~(y ,y -m) (m(xy)Bod(y )m(y-m)~od(x ) ,nor (x ) )dxd(X) (y ) d~o(X ), and

( (P iB)L ( f )~ ,n ) =

f f ( x ) g i ( x Y ) o ( y - l x - l , x ) ~ ( x y , y - l x - 1 ) ( L ( x y ) B o d ( y ) k ( y - 1 ) ~ d ( x ) , n o r ( x ) ) dxd(X)(y) d~o(X ).

One notes that

o (x , y )~ (y ,y -1) = ~ ( y - l x - l , x ) j ( x y , y - l x - Z ) .

Hence the fo l lowing estimate holds:

[ ( (L ( f )P iB - P iBL( f ) )~,n)1

[IBH f I f ( x ) l H~°d(x)II llnor(x)ll ; ] g i ( x Y ) - g i (Y) l dxd(X)(Y) duo(X)-

Since

j l f ( x ) t lt~°d(x)![ Nn~r(x)lt dvo (X)~ tlfi l i !t~11 llhlf,

we may use 3.4 ( i i ) to conclude that the r i gh t hand side goes to zero.

Hence L( f ) PB = (PB)L(f) and PB E ~ ' .

Q.E.D.

3.5. Remarks :

(a) R.Zimmer has introduced in [ 8 ~ , d e f i n i t i o n 4.1, the fo l lowing not ion of invar ian t

mean fo r (G,~). I t is a pos i t i ve un i ta l l i nea r map m from L~°(G,~) onto L~(GO,~)

sa t i s f y ing

( i ) m(h¢) = hm(¢) fo r ¢~L~(G,v) and h s Cc(GO ), where he(x) = ho r (x )¢ (x ) , and

( i i ) m(f¢) = fm(¢) f o r ¢ s L<(G,v), and f s Cc(G ), where f¢(x) = J f ( y ) ¢ ( y - l x ) d ~ r ( x ) ( y )

and where f@(u) = f (y)~od(y)dxU(y) fo r ¢ s L(GO,u). By a compactness argument, the

existence of an approximate invar ian t mean as in 3.4 gives the existence of an invar ian t

mean. The converse is probably t rue, but I don ' t have a cor rec t proof. I t can be shown

as in [82] and [83] , where the case of an ergodic act ion of a countable d isc re te group is

considered, that fo r a d iscre te groupoid G, the regular representat ion on u generates an

i n j e c t i v e yon Neumann algebra i f f there is an i nva r ian t mean fo r (G,u).

92

(b) R. Zimmer has also defined in [ 8 ~ , d e f i n i t i o n 1.4, an amenable ergodic group

act ion by a f ixed point property. This property is equivalent to the existence of

an invar ian t mean in the discrete case ( [ 8 ~ , 4.1) but the general case is unknown.

The de f i n i t i on of amenabil i ty given in 3.4 implies the f ixed point property, as a

standard averaging process shows.

(c) One can also use the approximate invar ian t mean of 3.4 to average cocycles and

get a vanishing theorem (cf . Johnson, [48], 2.5, page 32).

3.6. De f in i t i on : Let us say that G is measurewise amenable i f every quas i - invar ian t

measure on G O is amenable.

I f a l l the representations of C*(G,o) are obtained by in tegrat ion and i f G is

measurewise amenable, i t resul ts from 3.2 that C*(G,~) coincides with the reduced

C*-a lgebra Cr~d(G,~ ) and from 3.5 that i t is nuclear.

A s u f f i c i e n t condition for G to be measurewise amenable is the existence of a

net ( f i ) in Cc(G ) such that

( i ) the functions u ~ S I f i (x)12d~U(x) are uniformly bounded in the sup-norm ; and

( i i ) the functions x ~ f f i ( xY ) f i (Y )d~d (X ) ( y ) converge to I uni formly on any

compact subset of G. This condi t ion is also necessary in the case of a group (cf .

Dixmier [19], 13.5.2, page 260) ; but I do not know i f i t is true in general. Since

th is condi t ion is handy, I ca l l i t amenabi l i ty, although I don' t have any real j u s t i -

f i ca t i on for i t .

A question which arises is the amenabil i ty of C (G,~) in the sense of Johnson

( [48] , 5, page 60) ; in pa r t i cu la r , does the above condi t ion imply amenabil i ty ?

Let us now look at how amenabil i ty is preserved under some operations.

3.7. Proposit ion : Let U be a l oca l l y closed subset of the un i t space of G. I f G

is [measure wise]amenable, the reduction G U is [measure wise] amenable.

Proof : Suppose G amenable. Then there ex is ts a net ( f i ) in Cc(G ) such that f i * f i

converges to I uni formly on the compact sets of G and I f i . f ~ ( u ) I < M for sui table

M and any u. Let (h i ) be an approximate i den t i t y on Cc(U ), bounded in sup-norm. Then

93

g i ( x ) : h i o r ( x ) f i ( x ) h iod(x ) , x c G U

def ines a net in Cc(Gu) s a t i s f y i n g the required proper t ies . The proof of the other

statement is s i m i l a r . We note tha t any q u a s i - i n v a r i a n t on U is equ iva len t to the

r e s t r i c t i o n to U o f a quas i - i n va r i an t measure on G O , namely, a q u a s i - i n v a r i a n t measure

on U is equ iva len t to the r e s t r i c t i o n of the sa tu ra t ion [u] of ~ w i th respect to

G (1 .3 .7 ) . We denote the r e s t r i c t i o n of ~u to G U by ~ and the induced measure w i th

u d~(u). Then fo r E c U, respect to { ~ } by v U = %x U

[~](E) = 0 i f f

i f f

i f f

i f f

i f f

i f f

v(d-l(E)) = 0 ;

f o r u a.e. u, xU(d ' l (E ) ) = 0 ;

fo r p a.e. u, x~ (d - l (E ) ) = 0 (because E U and ~ l i ves

on U) ;

vu(d-l(E)) = 0 ;

vu l (d- l (E) ) - = 0 (because ~ is quas i - i n va r i an t ) ;

~ ( E ) = 0 .

Q.E.D.

3.8. Propos i t ion : Let G be a l o c a l l y compact groupoid w i th Haar system, l e t A be

a l o c a l l y compact group and c a continuous l -cocyc le in ZI(G,A). Let G(c) be t h e i r

skew product (1 .1 .6 ) .

( i ) I f G is [measurewise] amenable, then G(c) is [measurewise] amenable.

( i i ) I f A is amenable and G(c) is [measurewis~ amenable, then G is [measurewis~

amenable.

Proof : Let us reca l l the d e f i n i t i o n 1.1.6 of G(c) : G(c) = G × A wi th

( x , a ) ( y , a c ( x ) ) = ( xy ,a ) and (x ,a) - I = ( x - l , a c ( x ) ) . I t s u n i t space is G O × A. The

l o c a l l y compact groupoid G(c) has been def ined before 1.4.10. I f { u} is a Haar sys-

tem fo r G, a Haar system { u,a} fo r G(c) is given by

f f ( x , b ) d ~ u ' a ( x , b ) = ~ f (x ,a )d~U(x) .

Let us descr ibe q u a s i - i n v a r i a n t measures fo r G(c). Suppose tha t ~ is a q u a s i - i n v a r i a n t

measure on G O f o r G and {mu} a system of measures on A which is ~adequate (Bourbaki

[613.1) ( t h i s means tha t p = f ~ud_~(u) is wel l def ined) and which s a t i s f i e s

~d(x) ~ ~ r (x ) c(x) f o r ~ a . e , x, whe re~ i s the induced measure on ~. Then ~ = I ~ u dp(u)

94

is a quas i - invar ian t measure fo r G(c). Conversely, i f the quas i - invar ian t measure

can be d is in tegra ted along the f i r s t p ro jec t ion of G O × A, i t is of that form. For

the proof of th is fac t , we may assume that ~,~ and ~ a.e. mu are p robab i l i t y measures

and we may replace {~u} by equivalent p robab i l i t y measu'res. The measure v induced by

is of the form

S fdv = ~ f (x ,a )d~U(x)d~u(a)d~(u) ,

whi le - 1 is of the form

fdv -1 =~ f (x - l ,ac (x ) )d~U(x )dmu(a)d~(u) .

In pa r t i cu l a r , fo r any measurable set E in C, v(E x A) = ~(E) and ~-I(E × A) = ± - I (E ) .

This shows that ~ is quas i - invar ian t . Then, using the uniqueness of the d i s in teg ra t i on

of v along the f i r s t p ro jec t ion of G × A, one gets mr(x) ~ ~d(x) c(x-1) fo r v a . e . x .

Conversely, there is no problem checking that a measure ~ of the above form is quasi-

i nvar ian t .

( i ) Let u = ~ u d ~ ( u ) be a quas i - invar ian t measure fo r G(c) as above. I f ~ is

amenable, there ex is ts a net ( f i ) in Cc(G ) such that u -~ f i * f#(u) converges to 1

weakly . in L=(GO,~), and x ÷ f i * f~(x) converges to I w e a k l y * i n L~(G,~). Let (h i )

be an approximate i den t i t y fo r Cc(A ) with pointwise mu l t i p l i ca t i on and bounded in

sup-norm and def ine gi ~ Cc(G × A) by

g i ( x ,a ) = f i ( x ) h i ( a ).

The net (g i ) has the required proper t ies :

gi * g~(x,a) = ~g i ( xy ,a ) g i (Y ,ac (x ) ) d~d(X)(y)

= h i (a) hi (ac(x)) f i * f i (x) -

Let us check the convergence of (u,a) ÷ g i * gi (u 'a ) " The net is bounded in

L~(G 0 × A,u) , hence i t is enough to check the convergence against funct ions of the

form f (u )g (a ) where f ~ Cc(G° ) and g ~ Cc(A ).

We see that

f (u )g(a) gi * g i (u 'a) d~(u,a)

= ~ f ( u ) ( [ g ( a ) l h i ( a ) l 2 d~u(a)) f i * f ~ ( u ) d~(u) goes to

f (u ) g(a) du(u,a) ,

Since ~ g ( a ) l h i ( a ) l 2 dmu(a ) goes to ~g(a) dmu(a ) in LI(GO,~) and f i * f i (u) goes to

I in (L~(GO,u), weak.), The convergence of g i * g i (x 'a) is proved in the same fashion.

95

This shows that ~ is amenable. One proves in the same way that i f G is amenable,

then G(c) is amenable.

( i i ) We assume that A is amenable and ~(c) is

measurewise amenable. Let ~ be a quas i - invar iant measure on G O . Then ~ = L x ~,

where m is a r igh t Haar measure for A, is quas i - invar iant for G(c). Since G(c) is

amenable, there exis ts an approximate invar iant mean (g i ) , gi ~ O, gi ~ Cc(G × A)

such that (u,a) + ~ gi (x,a)d~U(x) converges to i in (L~(G 0 x A,~), weak*), and (x,a)

÷ } Ig i (xy ,a) - g i (Y,ac(x) l d~d(X)(y) converges to 0 in (L~(G x A,v) , weak* ) . The

group A also has an approximate invar iant mean (k i ) : k i ~ 0 k i e Cc(A ) such that

S kj(a)d~(a) = i , and b ~ f l k j ( a b ) - ku(a)I dm(a) converges to 0 uniformly on the

compact subsets of A. Let us define f i j e Cc(C) by f i j ( x ) = fg i ( x , a ) k j ( a )dm(a ) . I t

is not hard to check that the family of functions u ~ f f i j ( x ) d X U ( x ) is bounded in

L~(GO,~) and the family of functions x ~ J l f i j ( x y ) - f i j ( y ) Id~d(X)(Y) is bounded in

L~(G,~). We w i l l show that , given a neighborhood of 1 in (L~(GO,~), weak*),

V : {h ~L~GO,~) : l } (h(u) - i ) @k(U) d~(u)[ ! ek' k : l . . . . . m},

where @k c Cc(GO), ~k > O, k=l . . . . . m, and a neighborhood of 0 in (L ' (G,Z), weak.),

W : { f ~ L=(G,z) : I ~ f ( x ) ~ ( x ) d~(x)l ! n~ ~=i . . . . . n} ,

there ex is ts f i j such that u ~ f f i j dxu is in V and x ÷ ~ I f i j ( x y ) - f i j ( y ) i d xd (X ) (Y )

is in W. Let M be a bound for the norm of the functions (u,a) ÷ f g i ( x , a ) dxU(x) in

L~(G 0 x A). We can choose j such that , for every ~ = I . . . . . n,

I I ~ ( x ) l I k j (ac(x) -1) - k j (a) l d~(a) dz(x ) < o~/2M

from now on, j is kept f ixed. We observe that

#k(U) ( f f i j ( x ) dxU(x)) d~(u)

= f (g i (x ,a ) d~U(x)) ~k(U) k j (a) du(u,a)

goes t o } #k(U) ku(a ) d~(u,a) = f~k(U) d~(u) as i goes to ~. Hence for i su f f i c i en t l y

large, u ÷~ f i j d~U is in V. S imi la r ly , for i s u f f i c i e n t l y large, we w i l l have

(~ Ig i (xy ,a) - g i ( Y , a c ( x ) ) I d ~ d ( X ) ( y ) ) I ~ ( x ) I k j ( a ) d ~ ( x , a ) ~ n~ /2

= 1 , . . . , n .

Writ ing

f i j ( x y ) - f i j ( Y )

= ~(g i (xy ,a) - g i (Y ,ac (x ) ) )k j (a )d~(a ) + ~ g i ( Y , a ) ( k j ( a c ( x ) -1) - k j (a ) ) do(a),

96

we obtain the estimate

l ~ ( ~ I f i j ( x y ) - f i j ( y ) I d ~ d ( X ) ( y ) ) ¢~(x) QZ(x)]

_< ~(~Igi(xy,a) - g i (Y,ac(x)) idxd(X)(y) l ~ ( x ) I k j ( a ) dv(x,a)

+~ ]~ (x ) l ( Ig i (Y,a)dxd(X)(y)) I k j (ac (x ) -Z ) -k j (a ) l dv(x,a)

_< ngo

This shows that ~ is amenable. One proves in the same way that the amenability of

G(c) and A implies the amenabil ity of G.

Q.E.D.

Dual statements hold for the semi-direct product.

3.9~ Proposition : Let G be a loca l l y compact groupoid with Haar system, l e t A be

a loca l l y compact group acting continuously on G by automorphisms leaving the Haar

system invar iant and le t G x A be the i r semi-direct product (1,1.7).

( i ) I f A is amenable and ~ is [measurewise] amenable, then G × A is

~easurewise] amenable.

A is [measurewise] amenable then G is ( i i ) I f the semi-direct product G x

[measurewise] amenable.

Proof : Let us f i r s t define the semi-direct product as a loca l l y compact groupoid

with Haar system. We require the map from A x G into G sending (a,x) into s(a) x

to be continuous. Recall (1.1.7) that G x A is the groupoid G x A with (x ,a) (y ,b) =

(x(s(a)y) , ab) and (x,a) -1 = (s (a -1)x - l ,a -1 ) . I ts uni t space may be ident i f ied with

G O . The product topology makes i t into a loca l l y compact groupoid. We say that the

automorphism s of G leaves the Haar system invar iant i f s-~ u = ~s(u) ; in other words

~f (s(x) )d~S- l (U)(x) = ~f(x)d~U(x) for f c Cc(GU). I f {~u} is a Haar system for G

and ~ a l e f t Haar measure for A, then {~u x ~} is a Haar system for G × A. Let us

check l e f t invariance :

~f (x ,a)(y ,b))dxs(a-1)d(X)(y)d~(b)

= ~f(x(s(a)y),ab)d~s(a-Z)d(X)(y)da(b)

=~f(xy,ab)dxd(X)(y)dm(b) = ~ f(y,b)d~r(x)(y)dm(b).

Since the proof of this proposition is not much d i f fe ren t from the previous one

and does not involve any d i f f i c u l t y , we w i l l jus t indicate how the various approximate

97

means may be constructed.

( i ) Given ( f i ) such tha t f i * f*i -, " J" a . I on A, 1 in G and (h~) such tha t h~ * h* ÷ J

set

g i j ( x , a ) = f i ( s ( a - l ) x ) h j ( a ) .

• A. Then, there ex is ts a subnet such that g i j * g i j ÷ 1 in G ×

( i i ) Given an approximate i n v a r i a n t mean (g i ) f o r G × A, we can def ine an

approximate i n v a r i a n t mean fo r G by se t t i ng

f i ( x ) = f g i ( x , a ) d~(a).

q.E.D.

3.10. Example : A t ransformat ion group a r i s i ng from the act ion of an amenable group

is always amenable but the converse is not t rue. Let G be a second countable l o c a l l y

compact group and H a closed subgroup ; i t can be shown tha t the t ransformat ion group

H\G x G is amenable i f f H is amenable. Hence a homomorphic image of an amenable

groupoid is not necessar i ly amenable ; however, i t is probably t rue tha t the asymp-

t o t i c range (1 .4 .3) o f such a homomorphism is amenable (c f . Zimmer [ 8 ~ , 3 .3) .

In conc lus ion, l e t us ask some very basic quest ions.

( i ) Is a closed subgroupoid of a measurewise amenable groupoid also measurewise

amenable ? This is probably t rue but I can prove i t only in the case of an r - d i s c re te

groupoid. The proof uses 3.3 and 4.1.

( i i ) Does 3.9 hold fo r more general extensions 7

( i i i ) Is amenabi l i ty preserved under (the appropr ia te not ion of ) s i m i l a r i t y ?

4. The C* -A lgebra of an r -D isc re te Pr inc ipa l Groupoid

Reduced C* -a lgeb ras of r - d i s c re te p r inc ipa l groupoids are genera l i za t ions in a l l

essent ia l respects of the usual * -a lgebras of matr ices. They appear in a d iagonal ized

form. That i s , C*(G O) is a maximal abel ian subalgebra, the image of a unique condi-

t i ona l expectat ion. The elements of Cred(G,~ ) are matr ices over G, the diagonal

matrices are the elements of C*(G O) and the expectat ion map is eva luat ion on the

98

diagonal. The ideal s t ruc ture of Cred(G,~ ) is eas i l y described. Ideals correspond to

open inva r ian t subsets of the un i t space. Part of the representat ion theory may be

convenient ly expressed in terms of the groupoid. For example, the regular represen-

ta t ion on ~ is primary [resp. type I , I I or 111] i f f the measure ~ is ergodic [resp.

type I , I I or I I ~ . Such C*-algebras are character ized by the existence of a special

kind of maximal abel ian subalgebras, which, in accordance with [ 3 ~ , where a s im i l a r

notion is introduced in the context of von Neumann algebras, we ca l l Cartan subalgebras.

In the fo l lowing proposi t ion, we use the reduced norm II fired, which has been

defined in 2.8 and the sup-norm II II~.

4.1. Proposit ion : Let G be an r -d i sc re te groupoid with Haar system and l e t ~ be a

continuous 2-cocycle. Then, the fo l lowing i nequa l i t i es hold fo r any f e Cc(G,~ ) :

( i ) IlfII~ ~ IIfnred ; and

( i i ) fo r any u ~ G O , J l f l2d~u ~ Itftl~e d .

The proof resu l ts d i r e c t l y from the fo l lowing lemma.

Lemma : Let 8 and o be as above and l e t x be a point in G with d(x) = u. Consider the

representat ion L of Cc(G,o ) induced by the point mass at u (see 2.7). Let ~ and n

be the un i t vectors au and ax respec t ive ly in the space L2(G,~u) of the representat ion

L. Then for any f ~ Cc(G,~), f ( x ) = (L ( f )~ ,n ) and f ( y ) = L ( f )~ (y ) fo r any y c G u.

Proof : This is immediate since L is given by (L ( f )~ ,n ) =

f f ( Y z ) ~ ( z - I ) ~(Y) ~(YZ,Z-1)d~U(z)d~u(y ) (see 2.7). Note also tha t , since G is

r -d i sc re te , ~u is the counting measure on Gu(see 1.2.7) . For the proof of the propo-

s i t i o n , note that I f ( x ) l ~ I IL(f)I 1 II~II llnll < IIflIre d by d e f i n i t i o n 2.8, and so

~ I f ( y ) I 2d~u (y ) = IIk(f)~II2 ~ IIL(f)II 2 II~II 2 ~ IIfIIre d-

Q.E.D.

The in jec t ion j of Cc(G ) in to Co(G), the Banach space of continuous funct ions on

G which vanish at i n f i n i t y , extends to a norm decreasing l i nea r map j of Cred(G,~ )

into Co(G ) .

99

4.2. Proposi t ion : Let G be an r -d i sc re te groupoid wi th Haar system and ~ a continuous

2-cocycle. Then

( i ) the map j from C~ed(G,o ) to CO(G ) is one-to-one ( there fo re , the elements

of C * red(G,~) w i l l be viewed as funct ions on G) ;

( i i ) any a c C*red(G,o) sa t i s f i es IIaIL _ < llaIlred and II al 2 I I i - < I Ia l1~ed _ < llall~, where the norm II III has been def ined in 1.4 (llall I may be i n f i n i t e ) ; and

( i i i ) the operat ions in the * -a lgebra Cred(G,~ ) may be expressed in the same way as

in the same way as in the . -a lgeb ra Cc(G,~), e x p l i c i t l y

• ~(x,x-1 * a (x) = a(x - I ) ) , fo r a ~ Cred(G, ~ ,

a . b ( x ) = a(xy)b(y - I ) ~ (xy ,y -1)d~d(X) (y ) , fo r a,b cCred(G,~ ) , and

ha(x) = hor(x) a(x) , fo r h ~ C#G O) and a ~ C~ed(m,o ). Proof :

( i ) Let ~ be a quas i - invar ian t p robab i l i t y measure on G O. The regular represen-

ta t i on on p is rea l ized in standard form on L2(G,v -1) (see 1.10). This representat ion

is the GNS representat ion associated wi th the s ta te ~oP(f) = ~P( f )d~ = (L( f )~o,~o)

where P is the r e s t r i c t i o n map Cc(G,~ ) ÷ Cc(GO ) and #0 is the cha rac te r i s t i c funct ion

of G O , considered as a un i t vector in L2(G,~- I ) . In pa r t i cu l a r , ~0 is cyc l i c and

separat ing fo r the l e f t representat ion. We may wr i te L ( f )6 0 = f . @0 = ~ ( f ) fo r

f ~ Cc(G,o ) where j is the natural map from Cc(G ) in to L2(G v - l ) . We note that by

C ~ ~ _ • 4.1 l l j ( f ) l l ~ IIfIIred . Hence the equa l i t y remains true for a ~ red(G,~) L(a)# 0 = j ( a )

As ~(a) = j ( a ) - 1 a .e . , j ( a ) = 0 : > ~ ( a ) = 0 ~ > L(a) = O. Since the regular repre-

sentat ions form a f a i t h f u l fami ly of representat ions of C* red(G,a), a = O.

C* ( i i ) By con t inu i t y , the i nequa l i t i es of 4.1 s t i l l hold fo r a ~ red(G,o). The

inequa l i t y Iiailre d ~ IIaNI has been wr i t t en here fo r completeness.

( i i i ) I t su f f i ces to j u s t i f y the passage to the l i m i t in the expressions which are

va l id for f ~ Cc(G,o ). For example, suppose that fn ÷ a and gn ÷ b in Cred(G,o ) , wi th

fn,gn ~ Cc(G,~ ). Then fn * gn (x) = ~ fn(xy)gn (y -1 )~ (xy ' y -1 )d~d(x ) (y ) " Because of the

est imate II IL ~ II Ilre d, fn * gn (x) ÷ a * b ( x ) ( a * b denotes the product of a and b).

On the other hand, because of the estimate II II 2 ~ II fire d where II II 2 is the norm of

L2(G,~d(x)) , fn(X. ) ÷ a ( x . ) and gn ÷ b in L2(G,~d(x)) , hence the r i gh t hand side goes

100

to rj a ( xy )b (y -1 )~ (xy , y -1 )dxd (X ) ( y ) .

Q.E.D.

Remark : (Cf. [31,1Z]) . I t seems hard to charac ter ize the range of the map j . In the

case of the p r inc ipa l groupoid I x I , where I is a countable d iscre te space, t h i s

amounts to charac te r i z ing the matrices of compact operatorS. We may note tha t , in t h i s

case, the cond i t ions S l a ( x ) I 2 d~U(x) <_ IIall 2 are s a t i s f i e d by the matr ix o f any

bounded operator .

Let us study now the ideal s t ruc tu re of the reduced C*-algebra of an r - d i sc re te

p r inc ipa l groupoid. In fac t one can do a l i t t l e be t te r .

4.3. D e f i n i t i o n : Let us say tha t a l o c a l l y compact groupoid G is e s s e n t i a l l y p r i n c i -

pal when fo r every i nva r i an t closed subset F of i t s un i t space, the set o f u 's in F

whose iso t ropy group G(u) is reduced to {u} is dense in F.

4.4. Proposi t ion : Let G be an r -d i sc re te e s s e n t i a l l y p r inc ipa l groupoid wi th Haar

system. Then fo r any q u a s i - i n v a r i a n t measure ~, any ~- representa t ion L of G on

and any f c Cc(G), the f o l l ow ing i n e q u a l i t y holds :

sup I f ( u ) l < l lk ( f ) l l where F is the support o f u. ucF

Proof :

I t su f f i ces to prove the i n e q u a l i t y I f ( y ) I < l lL ( f ) I I fo r y E F such tha t G(u) = {y } .

Let (Vn) be a fundamental sequence of neighborhoods of u.

There ex is ts a sequence (~n) o f square- in tegrab le sect ions of the H i l be r t bundle of

the representat ion s a t i s f y i n g

Supp ~nC V n and f ] ~ n ( U ) l 2 d~(u) = I

We w i l l show tha t the sequence ( L ( f ) ( n , (n) tends to f ( u ) .

We f i r s t w r i t e f as a f i n i t e sum o f func t ions supported on compact open G-sets : m

f = ~ f i " f i = h. ×Si wi th h i a Cc(GO ) and S i c 9. 1

We use 1.7. ( * ) to compute ( L ( f i ) ~ n , (n) :

( L ( f i ) ( n ' ( n ) = SVnnV n S#1 h i ( u ) (L (u S i )~n(U.S i ) ,~n(U) ) D- I /2(u Si) d#(u)

I f y ~ y -S i , we have even tua l l y V n n v n -S i I = ~ and ( L ( f i ) ~ n, ~n ) = 0 .

I01

I f u = u . S i , the G-set S i meets G O . For n large enough, VnS i is contained in G O and

( k ( f i ) ~n ,~n ) = ~V n h i (u ) l l~n(U)l l2d~(u) tends to h i (u ).

Q.E.D.

Let G be an a r b i t r a r y l o c a l l y compact groupoid wi th Haar system. I ts reduced

C -algebra has a d i s t i ngu ished fami ly o f idea ls , def ined by i nva r i an t open subsets

o f G O . This is wel l known in the cases studied p rev ious ly (e.g. [86] , 2 .29) . Let us

in t roduce some no ta t ion . J(A) w i l l denote the l a t t i c e o f ideals o f the C* -a lgeb ra A.

O(G) w i l l denote the l a t t i c e of i n v a r i a n t open subsets o f the u n i t space o f the

groupoid G.

For U in O(G), Ic(U ) = { f ~ Cc(G,~ ) : f ( x ) = 0 i f x ~ G U }

and I(U) is the closure of Ic(U ) in C* (G,o) red

Lemma : Let X be a l o c a l l y compact space and Y be a normal open subspace. Then,

the c losure of { f c Cc(X ) : supp fcY} in the induc t i ve l i m i t topology of Cc(X) is

{ f ~ Cc(X ) : f ( x ) = 0 i f x ~ Y}.

Proof : One const ructs an approximate i d e n t i t y fo r Cc(Y ) as fo l l ows . "There are

increasing nets ( V ) and (V', ~ ) of r e l a t i v e l y compact open subsets o f Y such that

~ c V'm whi le uV~ = uV'm = Y and there are func t ions e c Cc(Y ) supported on

V'm such that em = 1 on Vm I f f c Cc(X ) and f ( x ) = 0 i f x # Y~ supp ( f e ) c Y

and f e ÷ f in the induc t i ve l i m i t topology, Moreover, the set { f E C (X) : f ( x ) = 0

i f x # Y} is c l e a r l y c losed.

4.5. Proposi t ion : Let G be a l o c a l l y compact groupoid w i th Haar system and l e t

be a cont inuous 2 cocycle.

( i ) I f U is an i n v a r i a n t open subset of G O and F is i t s complement, then I(U)

is an ideal of C* (G,a) which is isomorphic to C~ed(Gu,~) and such that the quot ien t red

is isomorphic to Cred(GF,~ ).

( i i ) I f u be a q u a s i - i n v a r i a n t measure of support F, the ideal I (U) , where U is

the complement o f F, is the kernel of the regu lar representat ion on ~.

( i i i ) The correspondence U ~ I(U) is a one-to-one order preserv ing map from

O(G) in to ~(Cred(G,~)).

102

Proof :

( i ) Using the invar iance of U, one eas i l y checks that I c ( U ) is a s e l f - a d j o i n t

two-sided ideal o f Cc(G,a ). Indeed, suppose f c Ic(U) and g c Cc(G,a), then

f * g (x) = J f ( y ) g ( y - L x ) a ( y , y - l x ) d~r (X) (y) .

I f x # G U and r (y) = r ( x ) , y # G U and f (y ) = O, hence f , g(x) = O. Therefore

i t s c losure in Cred(G,a ) is a closed ideal of C;ed(G,~ ). The map j from Cc(Gu,~ )

to Cc(G,a ) which extends a funct ion on G U by 0 outs ide o f G U is a *-homomorphism

and is isometr ic fo r the reduced norm. In fac t , i f we compose i t wi th the regular

representat ion on ~, where u is a quas i - i nva r i an t measure on G, we obta in the regu lar

representat ion on UU' the r e s t r i c t i o n of u to U. Conversely, i f ~ is a quasi-

i nva r ian t measure on U, i t can be viewed as a quas i - i nva r i an t measure ~ on G O and

Indp ( f ) = Ind~ ( j ( f ) ) fo r f c Cc(GU,~ ). Hence we have an isometr ic *-homomor-

phismfrom Cred(Gu, a) to Cred(G,a ). The lemma shows that i t s image is I (U).

The r e s t r i c t i o n map p from Cc(G,a ) onto Cc(GF,~) is a *-homomorphism. I f u is

a quas i - i nva r i an t measure on F, we view i t as a quas i - i nva r i an t measure on G O , say 2.

We have Indu (p ( f ) ) = Ind~ ( f ) fo r f c Cc(G,c ) . Hence p decreases the reduced norm and

extends to a*-homomorphism from C* red(G,~) onto C;ed(GF,a) I ts kernel I c l e a r l y con-

ta ins I (U). Let L be a representat ion o f Cred(G,a ) which vanishes on I (U). Ne def ine

L F on Cc(GF,~ ) by LF(f ) = L ( f ' ) where f ' ~ Cc(G,a ) and f ' i G F = f . This makes sense

because L vanishes on Ic(U ). The map L F is a representa t ion of Cc(GF,{ ) and s a t i s f i e s

LFO p ( f ) = L ( f ) . I f u I and ~2 are d i s j o i n t quas i - i nva r i an t measures on G O , Indu I and

Indu 2 are d i s j o i n t representat ions and Illnd~mV~2(f')H = Max(I I Indum(f ' ) l l , l l Indu2( f ' )H) .

This gives the estimate HLF(f)IJ<]JLII ljfl]red. Hence L F extends to a representat ion of

Cred(GF,a ) and L fac tors through p. Therefore I - I (U).

( i i ) I t su f f i ces to show that the regu lar representat ion on a quas i - i nva r i an t

measure u of support G O is a f a i t h f u l representat ion of C;ed(G,a ). Let M be a represen-

ta t ioH of G O . I t is weakly contained in the representat ion def ined by ~. Since the pro-

cess o f induct ion preserves weak containment, the kernel of IndM contains the kernel

of Indu.

( i i i ) This is c lea r .

Q.E.D.

103

We are ready to give the announced r e s u l t on the ideal s t ruc tu re of the reduced

C* -a l geb ra of a p r inc ipa l r - d i sc re te groupoid. I t is wel l known in the case of a

t ransformat ion group ( c o r o l l a r y 5.16 of [24] and theorem 5.15 of [86] ) . l , l i thout the

assumption of r -d isc re teness , the problem has recen t l y been solved, in the case of a

t ransformat ion group, by E.Gootman andJ.Rosenberg [38].

4.6. Proposi t ion : Let G be an r - d i s c re te e s s e n t i a l l y p r i nc ipa l groupoid w i th Haar

system and a a cont inous 2-cocycle. Then the correspondence U ~ I (U ) is an order pre-

serving b i j e c t i o n between the l a t t i c e Lg(G) of i n v a r i a n t open subsets o f G O and the

C* l a t t i c e J ( r e d ( G , a ) ) of idea ls o f the reduced C*-algebra Cred(G,a) *

Proof : Let L be a o- representa t ion of G l i

measure ~. Let P be the r e s t r i c t i o n map from

(comments before 2.7) tha t P is a cond i t i ona l

representat ion associated wi th the state uoP.

hence jIlnd~ ( f ) I I <l lL(f) l [ fo r any f c Cc(G ).

v ing on the q u a s i - i n v a r i a n t p r o b a b i l i t y

Cc(G,~ ) onto Cc(GO). We have seen

expectat ion and tha t Indu is the GNS

I t resu l t s from 4.4 tha t luoP(f) l ~ IIm(f) l I,

In p a r t i c u l a r , i f L is a representat ion of Cred(G,~ ), i t s kernel is contained in I(U)

where U is the complement of the support of ~. Since the reverse i nc lus ion is c lear ,

i t s kernel is p rec ise ly I (U) . Hence the map U ~ I(U) is onto. Q.E.D.

Our next task is to j u s t i f y the statement tha t the reduced C ~ -a lgebra of an

r - d i sc re te p r inc ipa l groupoid appears in a d iagonal ized form.

4.7. Proposi t ion : Let G be an r - d i sc re te groupoid w i th Haar system and a a con t i -

nuous 2-cocycle. Then

( i ) an element a of Cred(G,a ) commutes w i th every element of C*(G O) i f f i t

vanishes o f f the iso t ropy group bundle G' = {x E G : d(x) = r ( x ) } ; and

( i i ) C* (G O ) is a maximal subalgebra of C~ed(G,a) i f f G O is the i n t e r i o r o f G'.

Proof : Since G O is open, Cc(GO)is a subalgebra of Cc(G,~) and C~(G O) is a

subalgebra of Cred(G,a ). I t consis ts exac t l y o f those elements o f Cred(G,a) which

vanish o f f the u n i t space G O Let a E C~e C* (GO). . d(G,a) and h ~ Then ah(x) =

a(x)hod(x) and ha(x) = hor(x) a (x ) . I f a(x) = 0 fo r any x such tha t d(x) # r ( x ) ,

then a(x)hod(x) = ho r (x )a (x ) holds fo r every x in G. I f a(x) # 0 f o r some x such

that d(x) # r ( x ) , then there ex is ts h ~ C*(G O) such tha t hod(x) = I and hor(x) = O,

consequently, a(x)hod(x) # ho r ( x )a ( x ) , and so ah # ha. The asser t ion ( i i ) is an

immediate consequence of ( i ) . Q.E.D.

104

4.8. Proposi t ion : Let G be an r -d i sc re te p r inc ipa l groupoid wi th Haar system and

a continuous 2-cocycle. Then the r e s t r i c t i o n map P : C:ed(G,a) --~ C~( m 0 ) is the

unique cond i t iona l expectat ion onto C*(G O) and is f a i t h f u l .

Proof : The proof is i den t i ca l to a proof one would give in the case of matr ix

algebras. Note that , by 4.2, P is wel l def ined. There is no d i f f i c u l t y checking

P has a l l the proper t ies of an expectat ion map. To show uniqueness, we use the same

device as in 4.3 or 4 .5 .d. Let a ~ Cc(G,~ ) and suppose that suppa does not meet the

diagonal A of G O G O × G Q. There ex is ts a × - again, we view G as a subset of G O

f i n i t e cover of r(supp a) by open sets U i i=1 . . . . . n on G O such that U i × U i n supp a

= @ fo r i=1 . . . . . n. Let ( h i } be a p a r t i t i o n of un i t y subordinate to th is cover, wi th

n ~ h11/2a ~hi(u ) = 1 fo r u ~ r(supp a). Then a = ( ~ h i ) a and 0 = • h i1 /2 . I f Q is any l l

cond i t iona l expectat ion onto C ~(GO), then n n

o = Q( hil/2ahi Ij2) = i hil/2QIa)hi112= h QIa) n

= Q( # hia ) = Q(a) .

Since & is closed and open in G, an a r b i t r a r y a in Cc(G,~ ) may be wr i t t en a = a I + a 2

where a I is the r e s t r i c t i o n of a to & and su~a 2 does not meet A. Consequently, Q(a)

= a 1. This shows that Q agrees with P on Cc(G,~ ), hence on Cred(G,~ ). To see that P

is f a i t h f u l , note that i f a c Cred(G,~), then P(a ~ * a)(u) = J i a ( x - 1 ) I 2 d~U(x).

Hence i f P ( a * ~ a) = O, then a(x) = 0 fo r a l l x.

Q.E.D.

4.9. D e f i n i t i o n : Let A be a C * - a l g e b r a and B an abel ian sub C ~ -a l geb ra . We ca l l

normal izer of B ( in A) the inverse semi-group

~kC(B) = { a , p a r t i a l isometry of A : d (a ) , r ( a ) ~ B and a(Bd(a))a * = Br(a)} where d(a)

and r (a) denote the i n i t i a l and f i na l pro jec t ions of a. An element a m J~r(B) induces

an isomorphism s a : b ÷ aba ~ of Bd(a) onto Br(a) ; we also denote the corresponding

p a r t i a l homeomorphism of d(a) onto r (a) in the spectrum B of B by s a. The inverse

semi-group of p a r t i a l homeomorphisms of B of the form s a wi th a cAr (B) is ca l led the

ample semi-group of (B,A) (or of B when there is no ambiguity) and is denoted ~ ( B ) .

105

The semi-group of p a r t i a l isometr ies of B is denoted%(B).The un i t spaces ofJ~r(B), ~(B)

and ~(B) can a l l be i d e n t i f i e d wi th the Boolean algebra ~ o f pro jec t ions of B.

Remark : I f B is a maximal abel ian subalgebra o f a C* -a l geb ra A,

~%(B) ÷~°(B) ÷ g (B) ~ is an exact sequence o f inverse semi-groups (see 1.1.17). Indeed, i f s a is an

potent in ~(B) , then ab = ba fo r any b c B, hence a c JC(B) n B =cU~(B).

idem-

Recall that the ample semi-group of an r -d i sc re te groupoid is the semi-group

o f i t s compact-open G-sets. In the case o f a p r inc ipa l groupoid, G-sets are uniquely

determined by the p a r t i a l t ransformat ions they induce on the un i t space. Therefore

can be viewed as a semi-group of p a r t i a l homeomorphisms of the un i t space o f the pr in -

c ipa l groupoid.

4.10. Proposi t ion : Let G be an r -d i sc re te p r inc ipa l groupoid wi th Haar system and

l e t ~ be a continuous 2-cocycle. Then the ample semi-group ~(B) of the maximal abe l ian

subalgebra B = C*(G O ) of the C * - a l g e b r a A = C* (G,a) coincides wi th the ample red

semi-group ~ o f the groupoid.

Proof : We f i r s t show that ~ i s contained in ~(B). I f s is a compact-open G-set,

i t s c h a r a c t e r i s t i c funct ion Xs is a p a r t i a l isometry in Cc(G,o ) which normalizes B,

as an obvious computation shows : ×s * ×Z = r ( s ) , where r (s ) is i d e n t i f i e d wi th i t s

= * = h s fo r h c Bd(s), where cha rac te r i s t i c func t ion , Xs . ×s d(s) , and ×s* h , × s

hS(u) = h(u-s) i f u ~ r (s) and 0 i f u # d(s) . Hence Xs induces the p a r t i a l homeo-

morphism u ~ u.s.

homeomorphism i t i

As in p ropos i t ion

a . h . a * ( u

fo r h ~ Cc(d(s)).

I f y ~ G~(s) does

funct ion h e Cc(d

Hence a(y) = 0 i f y ~ s. Moreover, la (y ) I

Cc(G), s must be a compact open set of G.

Conversely, suppose that a is in J~C(B) and l e t s = s a the p a r t i a l

nduces on G O . We want to show that i t s graph is a compact open G-set.

2.9 o f [31, I l l , a simple computation shows that

) = I [a (Y) [ 2 hod(y)d~U(y)

By d e f i n i t i o n , th is equals h(u.s) fo r u e r(s).We f i x u e r ( s ) .

not belong to s, then d(y) # u.s and there ex is ts an non-negative

(s)) such that hod(y) = l and h(u.s) = 0 ; th is impl ies a(y) = O.

= 1 i f y belongs to s. Since a is in

Q.E.D.

106

C ~ A las t property of the pair ( red (G,~ ) , C*(GO)) needs to be interpreted in terms

of the groupoid. This is the notion of regular abelian subalgebra introduced in

Dixmier ~7] in the context of von Neumann algebras.

4.11. De f in i t i on : An abelian sub-* -a lgebra B of a C*-a lgebra A is said to be

regular i f the l i near span of the elements of the form ab, where a ~ J~r(B) and b ~ B,

is dense in A.

4.12. Proposit ion : Let G be an r -d iscrete groupoid with Haar system and a a

continuous 2-cocycle. Then C*(G O ) is a regular subalgebra of C* red(G,~) i f f G can be

covered with compact open G-sets.

Proof : I f G can be covered with compact G-sets, then one can, by using a pa r t i t i on

of the un i ty , wr i te any f c Cc(G,~) as a sum of funct ions supported on compact open

G-sets and a funct ion supported on the compact open G-set s may be wr i t ten under the

form X s . h where h c Cc(GO ). Conversely, i f the space of continuous funct ions

* G supported on compact open G-sets is dense in C red ( ,~ ) , they cannot a l l vanish at a

given point x of G, Consequently such point x is contained in a compact open G-set.

Q.E.D.

The properties of the subalgebra C*(GO), when G is an r -d iscrete pr inc ipal

groupoid, may be summarized by int roducing, as in d e f i n i t i o n 3.1 of ~ I , I ~ , the

notion of Cartan subalgebra. Recall that the ample semi-group of an r -d iscrete p r i nc i -

pal groupoid with Haar system has the property of acting r e l a t i v e l y f ree ly on the

un i t space (1.2.14), in the sense that the set of f ixed points of each of i t s

elements is open.

4.13. De f in i t i on : An abelian sub-*-algebra B of a C*-a lgebra w i l l be cal led a

Cartan subalgebra i f i t has the fo l lowing properties :

( i ) i t is maximal abelian ;

( i i ) i t is regular ;

( i i i ) i t s ample semi-group~(B) acts r e l a t i v e l y f ree ly on i t s spectrum B ; and

( i v ) the exact sequenceS~÷°d~(B) ÷~(B) ~ ~(B) ÷SSsplits in the sense that there

exists a section k for s sa t i s fy ing k(se) = k(s)e, k(es) = ek(s) and k(e) = e, for

every e in S~and s in ~(B) .

107

Question : Is ( i v ) independent of ( i ) - ( i i i ) ?

4.14. Proposit ion : Let G be an r -d iscre te pr inc ipa l groupoid admitt ing a cover of

compact open G-sets and l e t a be a continuous 2-cocycle. Then C* (GO)is a Cartan

subalgebra of C * (G,a). red

Proof : I t is maximal abelian by 4 . 7 . ( i i ) , regular by 4.12 and i t s ample semi-group

~, which is the ample semi-group of G by 4.10, acts r e l a t i v e l y f ree ly on G O by

1.2.13. A section for J~r(C*(GO)) + ~ i s given by k(s) = Xs where Xs is the

charac ter is t ic funct ion of the G-set s.

Q.E.D.

This proposit ion admits a converse.

4.15. Theorem : Let B be a Cartan subalgebra of a separable C*-algebra A.

( i ) There exists an r -d iscrete pr inc ipal groupoid G admitt ing a cover by compact

open G-sets, a continuous 2-cocycle ~ and a *-homomorphism ~ of C* (G,~) onto A which

carr ies f a i t h f u l l y C*(G O ) onto B and the ample semi-group of G onto the ample semi-

group of B.

( i i ) The groupoid G is unique up to isomorphism and the 2-cocycle a is unique up

to a coboundary.

( i i i ) I f G is amenable, the*-homomorphism ~ is an isomorphism and B is the image

of a unique condi t ional expectat ion, which is f a i t h f u l .

Proof :

( i ) The ample semi-group ~(B) of B is an inverse semi-group of par t ia l homeomor-

phisms of B, defined on compact open sets. By 4.13. ( i i i ) and 1.2.13, the pr inc ipal

groupoid G associated to i t has a structure of r -d iscre te groupoid wi th Haar system

such that B becomes i t s un i t space and ~(g) i t s ample semi-group. Let k be a section

for s as in 4.13 ( i v ) . By d e f i n i t i o n of s, i t sa t i s f ies the covariance property k( t )

* a t a k( t ) = for each t c ~(B) and each a eqJ~(B). Hence, the extension is compatible

with the action of ~(B) on qd~(B) (see 1.1.17). As in 1.1.17, th is extension is de f i -

ned by a 2-cocycle } E Z2(g(B) ,~L(B)) . As in 1.2.14, there exists a unique continuous

108

2-cocycle a s Z2(G,T) such that

a (s , t ) (u ) = a(us,ust)

for every s , t s ~ ( B ) . We t r y to define a map # of Cc(G,a ) into A by the formula n n

¢(~ h i Xsi ) = # h i k (s i ) where h i s Cc(GO ) c C* (G O ) = B and s i s~ (B ) . The map ¢ is n

well defined.First,n any element of Cc(G,a)may be wr i t ten ~l hi Xsi" Second, suppose

that ~ h i ×si = 0 : f ind d i s j o i n t compact open G-sets t j , j = 1 , . . . ,m such that each

s i may be expressed as an union of t j ' s . We may wr i te s i = .u s i j t j where s i j = 0 or I J

, = - = Z. s i j k ( t j ) and = ~ s i j × t j . The equal i ty and Otj = 0 l t j t j Then k(s i ) J ×s i J

) x t j = ! hi×si = !. = 0 for each j because the t . ' s Z.(~ ~ i jh i . 0 implies s i j h i l r ( t j ) , J j 1

are d i s j o i n t , l herefore Z hik(s i ) = Z.(~. s i j h i ) k ( t j ) = O. The same argument shows that 1 j l

¢ is one-to-one. We note that ¢ is a*-homomorphism. For i f s is a G-set, then

×~ = ~ ( s - l s ) * × s - l ,

~(×~ ) : ~ ( s - i , s ) * k(s - i )

= [k(s -1) k(s) k (s - l s ) *] ~ k(s - I )

= k ( s ) * = ¢(XS)* ;

and i f s and t are G-sets, then

X s . X t = ~(s,t) ×st

¢(X s . ×t) = ~(s,t) k(st)

= k(s) k ( t ) k ( s t ) * k ( s t ) = k(s) k ( t )

: @(×s) ¢(×t ). The map ¢ is continuous when Cc(G,~ ) has the inductive l im i t topology. Indeed, le t

( f i ) converge to f in the inductive l im i t topology ; multiplying by a f i n i t e part i-

tion of unity, we may assume that the f i ' s and f have their support contained in a

common compact open G-sets ; the assertion is now obvious. We may apply theorem 1.21

(or rather, i ts corollary 1.22) to conclude that ¢ is continuous for the C -norm

of Cc(G,a ). Since A is separable ; B is separable and ~(B) is countable, hence G is

second countable. We have already noted (1.3.28) that a second countable r-discrete

groupoid has suf f ic ient ly many non-singular G-sets. Thus ¢ extends to a *-homomorphism

of C (G,a) into A. I t is onto because i ts range contains the elements ab with

109

a ~ ~(B) and b e B and B is regu la r .

( i i ) The groupoid G is uniquely defined by B and i t s ample semi-group. The 2-cocy-

cle ~ is determined up to a coboundary by the extension

~÷%(B) o.~(8)+ ~ (8) ÷ ~ .

( i i i ) I f G is amenable, proposit ion 3.2 shows that C*(G,~) = Cr~d(G,~ ).

The kernel of ¢ is an ideal of Cred(G,o ) which intersects Cc(G,o ) t r i v i a l l y , hence i t

is t r i v i a l by 4.6. The las t assert ion resul ts from 4.8.

Q.E.D.

4.16. Remark : I t w i l l be given in 3.2.5 an example of a C * -a lgebra with two

maximal abelian sub C * -a lgebras, one of which is a Cartan subalgebra, the other

sa t i s f ies ( i ) ( i i ) but not ( i i i ) of 4.13 and is the image of a unique f a i t h f u l condi-

t ional expectation.

Let us conclude th is section by reca l l i ng some facts , due to P. Hahn, per t inent

to the regular representations of a pr inc ipa l groupoid. The case of an r -d iscre te

pr inc ipa l groupoid is studied in [ 3 ~ .

4.17. Proposit ion : (P.Hahn [ 4 ~ ) . Let G be a second countable l oca l l y compact groupoid

with Haar system, ~ a continuous 2-cocycle and u a quas i - invar iant measure on G O .

Then

( i ) the a-regular representation on ~ (defined in 1.8) is a factor representa-

t ion i f f v is ergodic,

( i i ) i t is of type I (resp. I I I , I I , I l l ) i f f ~ is of type I (resp. I I 1, I I ,

I l l ) (defined in 1.3.13).

Proof : The assertion ( i ) and part of the assertion ( i i ) resu l t from his theorem ~.1.

The rest resul ts from his theorems 5.4 and 5.5.

5. Automorphisms Groups, KMS States and Crossed Products

This section i l l u s t r a t e s the use of groupoids in the study of basic problems

110

for C* -a lgeb ras . As usual G denotes a l o c a l l y compact groupoid with Haar system {~u}

and ~ a continuous 2-cocycle. Here A denotes a l o c a l l y compact abel ian group wi th

dual group A. The value of the character ~ ~ A at a ~ A is w r i t t en (~,a) . Let

c ~ ZI(G,A) be a continuous one-cocycle. Def ine, fo r each ~ ~ A,

~ ( f ) ( x ) = ( ~ , c ( x ) ) f ( x ) fo r f c Cc(G,o ).

5.1. Propos i t ion : Let c ~ ZI(G,A) and ~ be as above. Then

( i

( i i

( i i i

words

of C

~ is an automorphism of Cc(G,~ ) ;

mg extends to an automorphism of C ~ (G,{) ;

(C * (G,~),A,m) is a C * -dynamical system (see 7.4.1 in [60 ] ) , in other

is a continuous homomorphism of A in to the group Aut(C ~ (G,o)) o f automorphisms

(G,~) equipped wi th the topology of pointwise convergence ; and

( i v ) ~ leaves C* (G O ) pointwise f i xed .

Proof :

( i ) This is a rou t ine v e r i f i c a t i o n .

( i i ) F i r s t , one notes that ~ is isometr ic with respect to the I[ 111 norm :

f l~(f)I(x)d~U(x) = I If(x)Id~U(x). Hence m~ is continuous wi th respect to the C * = -norm and so is ~ - I ~ - 1

-X- P ( i i i ) I t is c lear that m : A÷Aut(C (~,~)) is a group homomorphism. Let us check

i t s con t i nu i t y . I t su f f i ces to check tha t the map g -~ mgf is continuous f o r any

f c Cc(G,o ) and fo r the topology of the It iT I norm. Let K be the support of f . For any

> O, there ex is ts a neighborhood V o f ~ in A such that fo r n c V I ( n , c ( x ) ) - (~,c(x)) 1

< c fo r any x c K. Then ]l~nf - ~ f l11 , r sup l~nf - ~g d~ u _< U

Hence, II~ f - ~ f l l I _< ~Ilfll I .

( i v ) Clear.

Q.E.D.

In the case when A is the group of real numbers, one may def ine a l i n e a r map

on the domain D(~) = Cc(G,{ ) by

(~ f ) (x ) = i c ( x ) f ( x )

5.2. Propos i t ion. Let c e Z I ( G , ~ ) , and l e t m and 6 be as above. Then

111

( i ) Cc(G,a ) cons is ts of en t i r e ana l y t i c elements fo r m ; and

( i i ) ~ is a * - d e r i v a t i o n and a pregenerator f o r ~. (A reference fo r unbounded

der i va t ions on C* -a lgeb ras is [65] ) .

Proof : We note tha t f o r any continuous func t ion # on G and f ~ Cc(G,a ) ,

!l~ftl ~ ll~flli s (sup !¢(x) l ) l f f t l i x~K

where #f denotes the pointwise product and K is the support of f . Therefore, Cc(G,a )

is in the domain of the generator o f m and ~ is i t s r e s t r i c t i o n , and

a t f - f e i t c ( x ) - 1 i c ( x ) ] IIfl] I . I } - - ~ - 6f II ~ sup l

t K

The same argument shows that 6nf ex i s ts f o r any in teger n and

II~nfll ~ (sup I c ( x ) i ) n l l f l I l - K

This proves the f i r s t asser t ion . Also, 6 is c losable and i t s c losure generates an

automorphism group which can be nothing but ~. Hence, the c losure of a is the

generator of m.

Q.E.D.

Let us say tha t an automorphism group m~ of a C* -a l geb ra is inner i f there

ex is ts a group of un i t a r i es U~ in the m u l t i p l i e r algebra such tha t

( i ) m~(A) = U~A U~ ~ fo r any element A of the C * - a l g e b r a , and

( i i ) ~ ÷ U~ is continuous fo r the s t r i c t topology. (Recall tha t the s t r i c t

topology on the m u l t i p l i e r algebra is def ined by the semi-norms A ÷ IIABII and A ~ IIBAII

fo r B in the o r i g i n a l a lgebra) .

5.3. Proposi t ion : Let c ~ ZI(G,A) and l e t ~ be the associated automorphism group

of C * (~ ,a) .

( i ) I f c c BI(G,A) , then m is inner .

( i i ) I f G is r - d i s c r e t e , p r i nc ipa l and amenable, the converse holds.

Proof :

( i ) One f i r s t observes tha t any bounded continuous func t ion on G O def ines in

1t2

the obvious way an element of the m u l t i p l i e r algebra of C* (G ,~ ) . I f c(x) = bor(x) -

hod(x) where b is a continuous func t ion on G O , then fo r each ~ in A, U~(u) = (~,b(u))

def ines a un i t a ry element of the m u l t i p l i e r algebra and fo r f e Cc(G,{),

~E(f)(x) = (~,c(x))f(x) = (U~f U* ~ ) (x ) . The con t i nu i t y of ~ ~ U~ is checked as in

5.1 ( i i i ) .

( i i ) I f m~(A) = U~A U~ , then U~ commutes with every element of C* (G O ) and

hence is i t s e l f diagonal (see sect ion 4 - we have not considered the m u l t i p l i e r

algebra there, but i t s elements can also be viewed as continuous funct ions on G).

Therefore U~ is o f the form U~(u) = (E,b(u)) where b is a continuous func t ion on G O

and c(x) = bor(x) - bod(x).

Q.E.D.

As an example, l e t us i n t e rp re t the theorem 4.8 of the f i r s t chapter. We

assume that G is an r - d i sc re te , p r inc ipa l and amenable groupoid with compact un i t

space. By 4.6 of th is chapter, C* (G,~) is simple i f f G is minimal. Let c ~ zl(G,R)

and assume tha t c is bounded. This amounts to saying the associated de r i va t i on ~ is

bounded, or equ i va len t l y , that the associated automorphism group is norm continuous.

Then the range of c R(c) is compact and the asymptotic range R (c) is zero. The

theorem states that i f G is minimal and c bounded, then c is in BI(G,A). In other

words, i f C* (G,~) is simple and ~ bounded, then a is inner . This is a p a r t i c u l a r case

o f a wel l known resu l t o f Sakai ( [64 ] , 4 .1 .11) .

When G is r - d i s c re te , p r inc ipa l and amenable, the asymptotic range R (c) of a

cocycle c ~ ZI(G,A) can be i d e n t i f i e d as the Connes spectrum (see [60], 8.8.2) of

the associated automorphism group. This is h e u r i s t i c a l l y c lear when one compares both

d e f i n i t i o n s :

R(c) = nR(cu)

where U runs over a l l non-empty open sets in G O , c U is the r e s t r i c t i o n of c to GIU,

and R(c) is the closure o f c(G) ; whi le

r (~) = nSp(~lB )

where B runs over a l l ~ - i nva r l an t , hered i ta ry non-zero sub C* -a lgeb ras of C* (G,~),

and Sp(m) is the Arveson spectrum of m([60] 8 . 1 . 6 . ) . I t can be seen in our case tha t

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Sp(m) = R(c) and that fo r every open set U in G O , C * (GIU,~ ) may be viewed as an

m- invar ian t hered i ta ry subalgebra o f C* (G ,~ ) . (This may be done in a fashion

analogous to 4.4) . However, in order to avoid the e x p l i c i t determinat ion o f the

hered i ta ry subalgebras o f C * ( G , o ) , we w i l l use an a l t e rna te d e f i n i t i o n of r (~) ( [60 ] ,

8.11.8) which uses only the idea ls o f the cross-product algebra C ~(G,~) x A. This

w i l l be done in 5.8.

Given a cocycle c in Z I ( G , ~ ) and 5 s ~ , + ~ ] , t h e (c,~)KMS cond i t ion fo r a

measure u on G O has been def ined in 1.3.15. I t is t ime to j u s t i f y th is terminology.

We have seen how a one-parameter automorphism group m of C*(G,~) is associated to c.

On the other hand, composing ~ with the r e s t r i c t i o n map from Cc(G,~)onto Cc(GO),

one obtains a pos i t i ve l i n e a r func t iona l # = ¢~ on Cc(G,~ ). A pos i t i ve l i n e a r func-

t i ona l on Cc(G,~)continuous fo r the induct ive l i m i t top~)logy - an equ iva lent term is

"pos i t i ve type measure" - w i l l be ca l led here a weight on C (G,~). This does not agree

with the usual d e f i n i t i o n of a weight on a C ~ - a l g e b r a (see [12], page 61), because

Cc(G,o ) is not always a hered i ta ry subalgebra o f C~(G,o), but i t is convenient here.

I f G is r - d i sc re te and u a p r o b a b i l i t y measure, ¢ is a state. We note tha t , wi th

above nota t ions , ¢ is m- invar ian t since c vanishes on G O .

5.4. Proposi t ion : Let c c Z I ( G , R ) , B c [0, + ~] and ~ be a measure on G 0. The

automorphism group associated wi th c is denoted by ~ and the weight associated wi th

is denoted by 0- Then the fo l l ow ing proper t ies are equ iva len t :

( i the weight ¢ s a t i s f i e s the (m,6)KMS cond i t ion (see [60] 8.12.2 or [6~ 6.1) ;

and

( i i ) the measure ~ s a t i s f i e s the (c,B)KMS cond i t ion (1 .3 .15) . Moreover, i f G

is p r inc ipa l and ~ f i n i t e , any weight ¢ which s a t i s f i e s the (~,~)KMS cond i t ion ar ises

from a measure u on G O .

Proof : We f i r s t consider the case when # is f i n i t e . Replacing c by 6c, we may assume

that 6 = 1.

( i ) ~ > ( i i ) Since # is 1 - KMS fo r m, we f ind that fo r any f ,g c Cc(G,g ) and

any t c~

114

~[~t(~)*g] = ~ [ g . ~ t + i ( f ~ , because f is ana l y t i c fo r ~ ( 5 . 2 . ( i ) ) . Let us evaluate both expressions.

For the f i r s t ,

# [ a t ( f ) ~ g ] = I e i t c ( y ) f (Y) g ( y - l ) d v ( Y ) , where v = I ~Ud~(u)'

wh i le fo r the second,

@[g * m t+ i ( f ) ] = Ig (Y) e i ( t + i ) c ( y - 1 ) f ( y -1 ) dr (y)

= ~ e i t c ( y ) f ( y ) g(y-1) e-C(y) d -1 (y ) .

In p a r t i c u l a r , f o r any f c Cc(G )

f ( y ) d r (y ) = I f (Y) e-C(Y) d v - l ( Y ) ,

so tha t D = ~V ex is ts and is equal to e -c (v a.e. ) . dv -1

( i i ) : > ( i ) The same computation shows tha t , i f u is q u a s i - i n v a r i a n t w i th

dv - e -c , then - I

dv @[mt(f) ~ ~ = @[g ~ ~ t + i ( f ) ] fo r any f , g c Cc(G,o ).

Second, we consider the case when ~ is i n f i n i t e . The = - ~IS cond i t ion asserts

tha t fo r any f c Cc(G,~),

- i @ ( f ~ ~ 5 ( f ) ) Z O.

A f te r a computation, th i s becomes

I l f l 2 c dv - I > O, where v = I ~Udu(u).

Hence, ~ s a t i s f i e s the ~ - KMS cond i t ion i f f c is non-negative on the support of v - I ,

which is the inverse image under d o f the support of ~. But th i s is j u s t the ~ - KNS

cond i t ion fo r u , namely supp ~ c M i n ( c ) = {u ~ G O : CiG u ~ 0} .

F i n a l l y suppose tha t G is p r i n c i p a l , ~ f i n i t e and tha t the weight ~, correspon-

ding to the pos i t i ve type measure ~, s a t i s f i e s the (m,B) KMS cond i t ion . Then fo r any

f ,g ~ Cc(G,~ ) and any t ~ ~ , we have

~ [ ~ t ( f ) . g] : ~[g . ~ t+ i ( f~ . Using a l e f t approximate i d e n t i t y fo r Cc(G,~ ) endowed wi th the induc t i ve l i m i t topology,

one gets #(hg) = ~(gh) f o r g ~ Cc(G,{ ) and h ~ Cc(GO ). We want to show tha t the support

of u is contained in G O . Suppose tha t g E Cc(G ) and supp g n G O = @. Since G is p r i n -

c i p a l , supp g may be covered by open sets U such that d(U) n r(U) = @ . Using a par-

115

n

t i t i o n of the un i t y , we may wr i t e g = # gi wi th d(supp g i ) n r (suppgi) = 0. I f we

choose h e Cc(GO ) which takes the value 1 on d(supp g i ) and 0 on r(supp g i ) , we have

#(gi ) = #(hgi ) = #(gi h) = O, hence #(g) = O.

Q.E.D.

5.5. Remarks :

a. Since i t is important to determine a l l KMS weights of a group of automorphisms,

we give the fo l l ow ing complement fo r ~ = ~. Let # be a weight corresponding to a

pos i t i ve type measure u on G. Then one can show that # s a t i s f i e s the (m,#) KNS

cond i t ion only i f suppv c c-1(0) n d-1(Min(c~. In p a r t i c u l a r , i f Min(c) is reduced to

one element u,then c - I (0 ) n d-1(Min(c) ) is also reduced to {u} because Min(c) is

c - l ( o ) i n v a r i a n t (1.3.16 ( i v ) ) . Thus there is only one KMS weight at ~, namely, the

point mass at u.

b. Given an (~,#) KMS weight #, i t is natural to look at the GNS representat ion L

i t generates. I t is the representat ion induced by ~ in the sense of 2.7. I t acts

on L2(G,v -1) by l e f t convolu t ion. Let [ be the von Neumann algebra i t generates. There

ex is ts a unique normal s e m i - f i n i t e weight ~ on £ which extends # in the sense that

~( f ) = ~oL(f) f o r f ~ Cc(G,~), and there is a unique automorphism group ~ which

extends ~,

ato L ( f ) = Lo~t ( f ) fo r f ~ Cc(G,~ )-

Let H be the operator of m u l t i p l i c a t i o n by c on L2(G,v-1). Then ~ is given by

~t(A) = e i tH A e - i tH

The operator H is in te rp re ted as the energy operator in th is representat ion.

Let us consider the case ~ f i n i t e . We f i r s t assume ~ = i . The representa t ion

L is in standard form. I t is the regu lar representa t ion on ~ and appears as the l e f t

representa t ion of the genera l ized H i l b e r t algebra introduced in 1.10. In p a r t i c u l a r

is the modular group of the f a i t h f u l normal s e m i - f i n i t e weight ~. The r e l a t i o n between

the modular operator A, which is given by m u l t i p l i c a t i o n by the R-N d e r i v a t i v e D,

and the energy operator H is A = e -H. In the case when # is a r b i t r a r y but f i n i t e ,

we replace c by ~c and obta in the r e l a t i o n A = e -#H between A and H.

In the case B = ~, the representat ion L is no longer in standard form. The

116

~-KMS cond i t ion is p rec ise ly the requirement H > O. One says tha t a vector ~sL2(G,v -1)

-1 has zero energy i f ~(x) = 0 fo r v a.e. x such that c(x) = 0 and tha t ¢ is a physical

ground weight ( [65 ] , d e f i n i t i o n 5.2) i f the space of vectors of zero energy is one-

dimensional. A necessary cond i t ion is tha t u is a po in t mass. In f ac t , the po in t mass

at u s B in(c) def ines a physical ground weight i f f [u] n B in(c) = {u } , where [u]

is the o r b i t of u.

c. Suppose that c s B I ( G , ~ ) , c(x) = bor(x) - hod(x). Then we know tha t ~ is inner .

I t is implemented by the group of un i t a r i es Ut(u ) = e i t b ( u ) . I f we th ink of b as the

energy func t i on , the i n t e r p r e t a t i o n of B in(c) is c lear : B in(c) = (u s G O : the

r e s t r i c t i o n of b to [u] reaches i t s minimum at u}. In the general case, we w i l l ca l l

c the energy cocycle of the system.

Given a cocycle c s ZI(G,A), we have defined the C*-dynamical system (C*(G,a) ,A,~) .

Our l as t task in th i s sect ion is to i d e n t i f y the crossed product C*-algebra

C*(G,a) x A as the C*-algebra of the skew product, tha t i s , C (G*(c ) ,a ) . Let us

reca l l some notat ions and int roduce new ones : G is a l o c a l l y compact groupoid w i th

Haar system (k u) ; a i s a continuous 2-cocycle in Z2(G,~) ; A is a l o c a l l y compact

abel ian group, noted m u l t i p l i c a t i v e l y ; i t s dual group r = A w i l l be noted

m u l t i p l i c a t i v e l y too ; and c is a continuous 1-cocycle in ZI(G,A). The skew product

G(c) is the l o c a l l y compact groupoid obtained by de f in ing on G x A the m u l t i p l i c a t i o n

( x , a ) ( y , a c ( x ) ) = (xy ,a) and the inverse (x ,a) -1 = ( x - l , a c ( x ) ) . A composable pa i r

w i l l be w r i t t en (x , y ,a ) instead of ( ( x , a ) , ( y , a c ( x ) ) . The groupoid G(c) has the Haar

system (Xu,a = ku x 6a). A cocycle on G l i f t s to a cocycle on G(c), fo r example,

we def ine a ( x , y ,a ) = a ( x , y ) .

Let (E,F,m) be a Banach, -a lgebra dynamical system, tha t i s , E is a Banach

* -a lgebra , r a l o c a l l y compact group and ~ a continuous homomorphism of r in to Aut(E)

equipped wi th the topology of pointwise convergence. Recall tha t LI (F,E) is the space

of E-valued func t ions on r in tegrab le w i th respect to the Haar measure of F ( in our

case, ? is abe l ian ) . I t is made in to a Banach*-a lgebra wi th the operat ions :

f * g ( ~ ) = I f ( n ) ~n[g -1~) ]dn,

f * ( ~ ) = f ( ~ - l ) ,

and the norm l l f I l l= ] l l f ( ~ ) I l d ~. A covar ian t representat ion of the system on a H i l be r t

117

space JC consists of a continuous uni tary representation V of r on JC and a norm -

decreasing nondegenerate representation M of E such that V(~) M(e) V(~) = M[mc(e)].

A covariant representation (V,M) has an integrated form. Namely,

L(f) = f M [ f ( ~ V(~)d~

defines a non-degenerate representation of L on JC. Conversely, i f E has a bounded

approximate i den t i t y , any non-degenerate representation of LI(F,E) is an integrated

form and the correspondence is b i jec t i ve . Al l th is is well known and we refer to [20]

for fur ther deta i ls . I f E is a C*-a lgebra, the crossed product C*-algebra E x F

is the enveloping C* -algebra of LI(F,E).

Recall that we defined the norm II III on Cc(G,~) by

Itflli = max {sup ~ I f t d x u, sup I I f l d Z u ) U U

I t is a *-a lgebra norm on Cc(G,o). We denote the completion of Cc(G,~) in the norm

II III by LI(G,o). One annoying problem with th is Banach * -a lgebra is the exis tence of

a bounded approximate i den t i t y . I t can be established without d i f f i c u l t y in the r -

discrete case (take a bounded approximate iden t i t y for C*(GO)) and when G is a

transformation group (take the pointwise product hie i , where e i is the character is t ic

function of a symmetric neighborhood of the iden t i t y of the group, normalized for the

l e f t Haar measure, and e i a bounded approximate iden t i t y for C*(GO)), but I don't

know i f i t always exists in the general case. Note that, as a Banach space, LI(G(c),~)

is Co(A,LI(G,o)), the space of Ll(G,o)-valued continuous functions on A which vanish

at i n f i n i t y .

5.6. Lemma : Let E be a separable Banach space, F a l oca l l y compact abelian group

and ~ a continuous homomorphism of r into the group of isometries of E equipped with

the topology of pointwise convergence. Then

f ~ ~(a) = fF ~-iEf(~)] (~,a) de defines a norm-decreasing l inear map with dense range from LI(F,E) into CO(F,E).

Proof : Clearly, f (a) is well defined and llf(a)II ~ NfII. By the Lebesgue dominated

convergence, f is a continuous function from F to E. I f f is a decomposable element n

= f i ( ~ ) e i , where f i c L1(r) and of L I ( r ,E) , that i s , an element of the form f(~)

e i ~ E for i = 1, . . . . n, then f vanishes at i n f i n i t y . Since decomposable elements

are dense in L I ( r ,E ) , the map sends LI(F,E) into CO(F,E). We want to show that i t has

118

dense range. Note that the map 6, defined by 6( f ) (~) = ~ - l [ f ( ~ ) ] is an isometry

of L I ( r ,E ) . Therefore, i t suf f ices to consider the map f ÷ f , where f(a) =

I f (C) (~,a)d~. Since the Fourier transform from L l ( r ) into CO(r ) has dense range, every

decomposable element of CO(~,E ) l ies in the range closure of the map f ÷ f . Since

decomposable elements are dense in CO(~,E), the range closure is Co(r,E ).

O.E.D.

5.7. Theorem : Let G, o, A, c and m be as above. Assume that LI(G,a) has a bounded

approximate uni t . Then the crossed-product C*-a lgebra C*(G,o) x A is isomorphic

to the C*-a lgebra C* (G(c ) ,a ) of the skewproduct.

Proof : Since the automorphism ~ of Cc(G,a ), given by ~ ( f ) ( x ) = ( ~ , c ( x ) ) f ( x ) ,

preserves the [[ II I norm, i t extends to an automorphism of LI(G,a). The cont inu i ty

of ~ : r÷Au t (L l (G ,~ ) ) is established as in 5 . 1 . ( i i i ) . By 5.6, the map from LI(A,LI (G,a))

to Co(A,LI(G,a)) = LI(G(c),~) defined by f ÷ f (a) = f r ~ - l [ f ( ~ ) ] ( ~ ' a ) d ~ is norm-

decreasing and has dense range. I t is a straight forward computation to check that i t

is a * -a lgebra homomorphism. Let us jus t wr i te down the relevant formulas

f ( x ,a ) = ~ f ( x ,~ ) (~ ,ac (x ) ) d~

for f ,g ~ Cc(G x A )cL I (A ,L I (G ,~ ) ) ,

f * g (x,~) = I f f (y ,n ) g(y- lx ,n-Z~) (n ,c (y -Zx) )a(Y,y - lx )d~r (X) (y )dn

f * (x,~) = ~(x - I , ~-1) ~-(x,x-Z) (~ ,c(x) )

and for f ,g E C O(A,Cc(G))CLI(G(c) ,a) ,

f . g (x,a) = ] f ( y ,a ) g (y - l x ,ac (y ) ) a ( y , y - l x ) d~r(X)(y)

f * (x,a) = ~ (x - l , ac (x ) ) T ( x , x -1 ) .

Composing with the homomorphism of LI(G(c),~) into C* (G(c) ,~) , we obtain a (norm-

decreas ing)*-a lgebra homomorphism ~ from LI (A,LI (G,a)) into C* (G(c ) ,a ) which has

dense range. I f L is a (non-degenerate) representat ion of C* (G(c) ,a) , Lo~ is a non-

degenerate representat ion of L I (A,L I (G,~) ) . There exis ts a covariant representat ion

(V,M) of (A, LI(G,a), of which Lo~ is the integrated form. By de f in i t i on of

C (G,a), M decreases i t s C*-norm and we obtain the estimate

IILo~(f)ll = l l~M[f(~)Iv(~)d~ll

~llf(~) 11d~ = Nfll I

119

where ]] I{ I is the norm of LI(A,C* (G,a)). Therefore, LoT extends to a representation

of L I (A ,C*(G,o) ) and ipso facto to a representation of i t s enveloping C*-a lgebra

C*(G,o)×m A. We have I]Lo~(f) II # [IfI[, where [I IIis the norm of C*(G,o) x A. We conclude

that ll~(f) II ~ Ilfll and that ~ extends to a *-homomorphism from C * ( G , o ) x A to

C* (G(c ) ,o ) . I t is onto, because i t s range is dense and closed. Let us show that i t

is one-to-one, or, equ iva lent ly , isometr ic. Let L be the representation of

C*(G,o) x A induced by the representation M of C* (G,~). We w i l l assume that M is

the integrated form (cf . theorem 1.20) of the {-representat ion (~,T~,M) of G. Let

H = r(~) (the space of square integrable sections o f ~ ) be i t s representation space.

By de f i n i t i on , L acts on L2(A;H) by

k( f )~(y) = f r M [ ~ y - l ( f ( ~ ) ) ] ~(~-Iy)d~

where f ~ L I (A;C*(G,~) ) and @ cL2(A;H). Let us consider the fol lowing o-representa-

t ion (~xx~JC,[) of G(c) : k is the Haar measure of A (we have observed in 3.8

that M×kis quas i - invar ian t ) , Jgu, a =~(Ju ; and [ (x ,a) : JC(d(x) ,ac(x)) ÷JC(r(x) ,a)

is given by [ ( x ,a ) = M(x). I ts integrated form acts on F(;}C) by

[(f)@(u,a) = f f ( x , a ) [ (x ,a)@(d(x) ,ac(x)) D- l /2(x)dxU(x) ,

for f ~ Cc(G(c),o ), @~F(JC), where D is the modular funct ion of ~. We may iden t i f y

F(JC) with L2(A,H) in an obvious fashion, where H = F(~), and we may define the

Fourier transform ~-from L2(A,H) to L2(A;H) by ~-@(a) = f@(y)(y ,a) dy. Of course,

~ - i s an isometry. I t is then a straightforward compu~tion to check that

~ok( f ) = [ ( f ) o~-

for any f ~ Cc(A x G), where f = ~( f ) . The relevant formulas are

L( f )~(u ,y) = f f ( y , c ( x ) ) f (x ,~) M(x)~(d(x) ,~- Iy) D-1/2(x) d~U(x)d~,

[(f)@(u,a) = f f ( x , a ) M(x)@(d(x),ac(x)) D -1/2 (x) d~U(x) ,

f (x ,a ) = f f ( x , ~ ) (~,ac(x)) dg, and

~-@(u,a) : f@(u ,y ) (~,a) dy .

This shows that IIL(f)H ~ II~(f)II for every f ~ L I (A ,C*(G,o) ) and every induced

representation L. Since A is abel ian, the reduced norm on LI(A,C* (G,~)) coincides

with the C*-norm ( [70] , proposi t ion 2.2). Hence llfll ~ II~(f)]I.

Q.E.D.

120

5.8. Coro l la ry : Let G be an r - d i s c re te amenable p r i nc ipa l groupoid w i th Haar system,

a E Z2(G,T), A a l o c a l l y compact abel ian group and c E ZI(G,A). Then the asymptot ic

range R (c) of c coinc ides w i th the Connes spectrum r(m) of the corresponding

automorphism group m on C*(G,~).

Proof : We i d e n t i f y the crossed product C* -a lgeb ra C* (G,o ) × A and C* (G(c),~) .

The canonical act ion of A on the skew product G(c), s (a ) (x ,b ) = (x ,ab) , def ines an

act ion on A on C * ( G ( c ) , ~ ) , Ba( f ) ( x ,b ) = f ( x , a - l b ) . Thus C*(G(c) ,~) ,A,~) is nothing

but the dual system of (G (G,~),A,m). The Connes spectrum r(m) can be character ized

as ( [60 ] , 8.11.8) r(m) = {a c A : J n Ba(J ) # {O} for every non-zero ideal J of

C (G(c ) ,o ) } . Using the correspondence 4.6 between ideals o f C*(G(c),~) and i n v a r i a n t

open subsets o f the un i t space o f G(c), the amenabi l i ty o f G(c) and 1.4.10, one gets

the conclus ion. O.E.D.

5.9. Remark : We have r es t r i c t ed our a t t en t i on to automorphism groups of C*(G,~)

which stem from a cocycle c c ZI(G,A). Another kind of automorphism group which

leaves C* (G O ) i nva r i an t is given by a cont inuous act ion of a group A by automorphisms

of G leav ing the Haar system i nva r i an t and a s im i l a r study can be done.

CHAPTER I I I

SOME EXAMPLES

We shall give here two kinds of examples of r -d iscrete groupoids with Haar system.

v Our f i r s t example resul ts from the observation by S t ra t i l ~ and Voiculescu ( [69] , ch. I ,

§ I , page 3) that approximately f in i te-d imensional C * -a lgebras ( fo r short AFC*-alge-

bras) could be diagonalized. This fact had already been used in a par t i cu la r case be o

Garding and Wightman in [34] to construct i n f i n i t e l y many non-equivalent i r reduc ib le

representations of the anticommutation re la t ions . In the terminology of 2.4.13, th is

can be rephrased by saying that AF C~-algebras have Cartan subalgebras. Thus, an

AF C*-a lgebra is the C*-a lgebra of an r -d iscre te pr inc ipa l groupoid. The groupoids

which arise in that fashion (we cal l them AF) are studied in the f i r s t section. They

have also been considered, in a form where the emphasis was on the ample group rather

than on the groupoid, by Krieger in [52]. Our second example is given by the C* -a lge -

bras generated by isometries introduced and studied by Cuntz in [15]. We show that these

C*-algebras may be wr i t ten as groupoid C*-a lgebras. The corresponding groupoids, which

are described in the second section and which we ca l l O n , are not p r inc ipa l . In both

cases, the descr ipt ion of the C*-a lgebra in terms of a groupoid is used to discuss

the existence of KMS-states with respect to some automorphism groups.

1. Approximately F in i te Groupoids.

The simplest examples of r -d iscrete pr inc ipal groupoids are, on one hand, the

l oca l l y compact spaces (corresponding to the equivalence re la t ion u ~ v i f f u = v)

122

and, on the o ther , the t r a n s i t i v e p r inc ipa l groupoids on a set o f n elements, where

n = 1,2 . . . . ~ (corresponding to the equivalence r e l a t i o n u ~ v fo r every u and v) wi th

the d iscre te topology. By means o f elementary operat ions, we may combine them to obta in

other examples.

The product o f two groupoids is def ined in the obvious fashion. I f the groupoids

are t opo log i ca l , then the product is given the product topology and i f each o f the

groupoids is endowed wi th a Haar system, the product is given the product Haar system ;

u. (Ul,U2) u I u 2 e x p l i c i t l y i f {~i I } is a Haar system fo r Gi, i = 1,2, then {~ = ~1 x ~2 } is

a Haar system fo r G 1 x G2"

Another operat ion makes sense in the category o f groupoids ; t h i s is the d i s j o i n t

union. Let G i be a groupoid, with i = 1,2 ; then def ine G = G 1 • G 2 as the set-

t heo re t i ca l d i s j o i n t union o f G 1 and G 2 wi th the groupoid s t ruc ture given by the ru les

"x and y are composable in G i f f they belong to the same G i and are composable in G i

and t h e i r product in G is equal to t h e i r product in Gi" and " i f x belongs to G i , i t s

inverse in G is equal to i t s inverse in Gi". I f the groupoids G i are t opo log i ca l , u.

then t h e i r d i s j o i n t union is given the d i s j o i n t union topology and i f {~i I } is a

Haar system fo r Gi, i = 1,2, then {~u}, where ~u = I~ i f u ~ Gi O, is a Haar system

fo r G. One can def ine in a s im i l a r fashion the d i s j o i n t union o f a sequence o f grou-

poids.

A l a s t operat ion which we need here is the induct ive l i m i t . We give here a

res t r i c t ed d e f i n i t i o n , s u f f i c i e n t fo r our purposes. Suppose that the groupoid G is

the union o f an increasing sequence o f subgroupoids Gn, which a l l have the same un i t

space as G ; then we say that G is the induct ive l i m i t o f the sequence (Gn). I f G

is t opo log i ca l , we requi re that G n be an open subgroupoid o f G. I f {~u} is a Haar

u is the system fo r G, we consider the Haar system { ~ } on G n such that ~n

r e s t r i c t i o n o f u to r n l ( u ) . Conversely, suppose that the G n s are topo log ica l

groupoids such that G n is open in Gn+ 1 and i t s topology is the topology induced from

Gn+ I . Then, the induct ive l i m i t topology, where a set V is open i f f V n G n is open

in G n fo r every n makes G in to a topo log ica l groupoid I f the ~n s are l o c a l l y compact,

u then so is G. F i n a l l y , i f each G n has a Haar system {~n } and i f these measures are

u l (u ) u is the r e s t r i c t i o n o f ~n+l to r~ then there compatible, in the sense that ~n

123

ex is ts a unique Haar system { u} such that ~u is the r e s t r i c t i o n o f ~u to rn l (u ) n

Let us note that these operat ions preserve amenabi l i t y ( d e f i n i t i o n 2.3.6) . Let

us show, fo r example, that the induct ive l i m i t G of a sequence (Gn) of amenable

groupoids is amenable. Let K be a compact subset o f G and E a pos i t i ve number. Since

I the G n s are open, K is contained in some G n. Since G n is amenable, there ex is ts

2 u f ~ Cc(Gn) such that I f * * f ( x ) - 11 ~ ~ fo r x E K (and j I f ( x ) I dA n bounded by 2).

Then f E Cc(g ) and s a t i s f i e s the same cond i t ion in G.

1.1. D e f i n i t i o n : Let G be an r -d i sc re te groupoid. We say that G is an elementary

groupoid o f type n (n = 1,2 . . . . . ~) i f i t is isomorphic to the product o f a second

countable l o c a l l y compact space and o f a t r a n s i t i v e p r inc ipa l groupoid on a set of

n elements.

We say that G is an elementary groupoid i f i t is the d i s j o i n t union o f a sequen-

ce o f elementary groupoids o f G i o f type n i .

We say that G is an approximately elementary (AE! groupoid i f i t is the induct ive

l i m i t of a sequence of elementary groupoids.

We say that G is an approximately f i n i t e (AF) groupoid i f i t is approximately

elementary and i t s un i t space is t o t a l l y disconnected.

1.2. Remarks : A l l these groupoids are p r inc ipa l and amenable since these proper t ies

are preserved under product, d i s j o i n t union and induc t ive l i m i t . They have the coun-

t ing measures as Haar system.

The o rb i t s o f an elementary groupoid of type n have the same c a r d i n a l i t y n. How-

ever there ex i s t r - d i sc re te p r inc ipa l groupoids, a l l o rb i t s o f which have the same

c a r d i n a l i t y n, which are not elementary o f type n. An example is given by the equi-

valence r e l a t i o n on the c i r c l e which i d e n t i f i e s two points l y ing on the same diameter.

The un i t space of t h i s groupoid is connected, whi le the un i t space of an elementary

groupoid o f type 2 has a t leas t two components.

The terminology o f elementary groupoid does not agree with the d e f i n i t i o n

(4.1.1) in [ 1 ~ ) o f an elementary C * - a l g e b r a . Only t r a n s i t i v e p r inc ipa l groupoids

give elementary C* -a lgebras.

124

1.3. Propos i t ion :

( i ) Let G be an elementary groupoid. Then, f o r every G-module bundle A (not

necessar i l y a b e l i a n ) , every cocycle c c ZI(G,A) is i nner , ( t ha t i s , is a coboundary).

( i i ) Let G be an approx imate ly e lementary groupoid. Then, f o r every G-module

bundle A (not necessar i l y a b e l i a n ) , every cocycle c E ZI(G,A) is approx imate ly inner

in the sense tha t i t can be approximated by coboundaries un i fo rm ly on the compact

subsets o f G.

( i i i ) Let G be an approx imate ly elementary groupoid. Then, f o r every abe l ian G-

module bundle A and every n ~ 2, Hn(G,A) = 0.

Proof :

( i ) We w i l l show tha t an elementary groupoid is (cont inuous ly ) s i m i l a r to a

l o c a l l y compact space. Since a l o c a l l y compact space (as a groupoid) has t r i v i a l co-

homology, t h i s w i l l prove the asse r t i on . I t su f f i ces to consider the case of an

elementary groupoid o f type n, o f the form G = X x I , where X is a l o c a l l y compact n

space and I n the t r a n s i t i v e groupoid on {1 . . . . . n} . Then, a s i m i l a r i t y between

G and X is given by

: X × I n ÷ X and ~ : X ÷ X x I n

( x , ( i , j ) ) ~ x x ~ ( x , ( 1 , 1 ) )

because #o~ = id X and ~ o ¢ ( x , ( i , j ) ) = e ( x , i ) i d G ( X , ( i , j ) ) e ( x , j ) -1 where e is the map

X x {1 . . . . . n} ÷ X x I n

( x , i ) ~ ( x , ( i , i ) ) ,

( i i ) Let G be the i nduc t i ve l i m i t o f a sequence o f elementary groupoids G n

and l e t c c ZI (G,A) . By ( i ) , the r e s t r i c t i o n ClG n o f c to G n is a coboundary on G n,

hence may be extended to a coboundary c n on G. Since every compact subset o f G is

conta ined in some G n, (Cn) converges to c un i fo rm ly on the compact subsets o f G.

( i i i ) Wri te G as increas ing union o f a sequence of elementary groupoids G i ,

Let ~ e Zn(G,A), w i th n ~ 2. I t s r e s t r i c t i o n to G i , ~ i ' belongs to Zn(Gi ,A) .

However Zn(Gi,A) = (0) f o r n ~ 2, since Zm(Gi,A) = Bm(Gi,A) f o r m > 1. Thus o = O.

Q.E.D.

125

An essen t ia l fea tu re o f an approx imate ly elementary groupoid G is tha t i t has

(c,~) KMS measures fo r every c c ZI(G,~) and every B E [ -~ ,+~ ] , provided tha t i t s

u n i t space is compact.

1.4. Lemma : Let G be a l o c a l l y compact groupoid w i th Haar system and l e t c be a

coboundary in BI(G,J~).

( i ) I f G O is compact, then (c,~) KMS p r o b a b i l i t y measures e x i s t .

( i i ) I f there is a (c ,~) KMS measure fo r some ~ c ~ , then there are ( c ,B ' ) KMS

measures fo r every 8' E ~ .

( i i i ) I f G O is compact and i f there is a (c,B) KMS p r o b a b i l i t y measure f o r some

B e ~ , then there are ( c ,B ' ) KMS p r o b a b i l i t y measures fo r every 8' E [ -~ ,+~ ] .

Proof : Let us w r i t e c(x) = hor (x) - hod(x) where h is a continuous func t ion on G O .

( i ) The set Minh o f the points o f G O where h reaches i t s minimum is non-empty

and conta ined in Minc. The point-mass a t such a po in t is a (c ,=) KMS p r o b a b i l i t y

measure.

( i i ) I f p is a (c,B) KMS measure, then fo r every 8' e ~ , the measure ~' given by

d~ ' (u ) : e x p [ - ( ~ ' - B ) h(u) ] d~(u)

is a ( c , # ' ) KMS measure. For, i f v' = I ~Udu ' (u ) :

d r ' d~l -1 (x) = e x p [ - ( B ' - ~ ) h o r ( x ) ] (x) exp [ ( ~ - ~ ' ) h o d ( x ) ] d~' d~

= exp [ -Bc (x ) ] .

( i i i ) I f u, as above, is f i n i t e and i f G O is compact, u' is a lso f i n i t e .

Q.E.D.

1.5. Propos i t ion : Let G be an approx imate ly elementary groupoid w i th compact un i t

space. Then i t admits (c,B) KMS p r o b a b i l i t y measures fo r every c E ZI(G,~) and every

B E [ -~ ,+~ ] .

Proof : Since elementary groupoi~s w i th compact un i t space have f i n i t e i n v a r i a n t

measures, they have (c,B) KMS p r o b a b i l i t y measures fo r every c and every 8. Fix

E~ andc c Z I (G,~) . Wri te G as the i nduc t i ve l i m i t o f a sequence (Gn) o f e lementary

groupoids and l e t c be the r e s t r i c t i o n of c to G . For each n, there ex i s t s a n n

126

-6c n p r o b a b i l i t y measure ~n whose modular funct ion wi th respect to G n is e Let

be a l i m i t po in t o f the (Un)'S fo r the weak , - t o p o l o g y o f the dual o f the space

-1 ÷ v - I fo r the weak of continuous funct ions on G O . I f Un ÷ u' then Vn ÷ ~ and ~n

* - t opo logy o f the dual o f Cc(G ). Therefore, for every f E Cc(G ),

Jfdv - I = l im J f d ~ n l = l ira J fe6Cnd~n = J fe~Cd~.

This shows tha t the modular funct ion o f ~ ex is ts and is e -~c.

The statement about i n f i n i t e 6 resu l ts from 1.3.17.

Q.E.D.

1.6. Example : The Is ing model.

The points o f Z = Z v are the s i tes o f a crysta l l a t t i c e of dimension v,

where v is an in teger . Each s i te has a spin up ( - I ) or down ( - i ) . A con f igura t ion o f

the l a t t i c e is given by a funct ion u o f Z in to { - 1 , + I } . The space o f con f igu ra t ion

{ - 1 , + I } Z is given the product topology ; i t w i l l be the un i t space G O of the groupoid.

Two conf igura t ions are equ iva lent i f f they d i f f e r at most f i n i t e l y many s i tes .

The corresponding p r inc ipa l groupoid is noted G. We choose an increasing sequence

(Zn) o f f i n i t e subsets o f the l a t t i c e such that Z = u Z n and def ine the subgroupoid

G n by the equivalence r e l a t i o n : "two conf igura t ions are equ iva lent i f they agree

outside Z n . Then G n is an elementary groupoid of the form { - I , + I } Z\Zn × l [Zn] and

G =uGn. We give to G the induct ive l i m i t topology. Thus G is an AF groupoid.

The dynamics o f the system are described by the fo l l ow ing energy cocycle

c ~ Z I (G ,R) given by the expression

c(u,v) = .~. J ( i , j ) { ( i - u iu j ) - (1 - v i v j ) } , l~J

where J depends on the nature o f the i n t e rac t i on . The sum is in fac t f i n i t e since

there are f i n i t e l y many non zero terms.

From 1.5, the system has I~MS states fo r every ~. The ground states are the

G o measures which l i v e on {u ~ : uiu j = i whenever J ( i , j ) # 0}. In p a r t i c u l a r , the

conf igura t ions (u i = +i fo r every i ) and (u i = -1 fo r every i ) are physical ground

states.

Some resu l t s , depending on ~ and on J, are known above the existence o f d i s t i n c t

KMS states at a given B.The parameter B is in te rp re ted as the inverse temperature

127

and ~4S states are equ i l ib r ium states. Coexistence of d i s t i n c t KMS states means the

existence of several "phases". I f the l a t t i c e were f i n i t e , G would be f i n i t e , c inner

and there would be one and only one KMS state for every 6. The interested reader w i l l

f ind a review of these resul ts as well as a bibl iography in the A.M.S. a r t i c l e by

J. Fr~hl ich [33].

We turn now to the propert ies of the skew-product G(c) where G is approximately

elementary (or f i n i t e ) .

1.7. Proposit ion : Let G be a l oca l l y compact groupoid, A a l oca l l y compact group

and c a cocycle in ZI(G,A).

( i ) I f G is approximately elementary, then the skew product G(c) of G by c is

approximately elementary.

( i i ) I f G is approximately f i n i t e and A is t o t a l l y disconnected, then G(c) is

approximately f i n i t e .

Proof :

( i ) I f c is a coboundary, c(x) = bor(x) (bod(x)) - I Then, the map from G x A

to G(c) sending (x,a) to (x, a(bor(x)) -1) is an isomorphism of groupoids, when G × A

is given the product st ructure and where A is viewed as a l oca l l y compact space.

Therefore, i f G is elementary, G(c) is also elementary for every c c ZI(G,A).

Suppose now that G = uGn with G n elementary. Let c ~ Z 1 (G,A) and l e t c n be

i t s r es t r i c t i on to G n. Then G(c) = UGn(Cn) and Gn(Cn) is elementary. Thus, by de f i n i -

t ion , G(c) is approximately elementary.

( i i ) From the f i r s t part , we know that G(c) is approximately elementary. Moreover

i t s un i t space G O × A is t o t a l l y disconnected. Hence i t is approximately f i n i t e .

Q.E.D.

Remark : This las t proposit ion gives a par t ia l answer to a question Bra t te l i asks

in [ 4 (problem 2, page 35). I f (~t,G,~) is a C* -dynamical system with.,{ AF and G

compact, is the crossed product a lgebra~x G necessari ly AF ? This is so i f G is

abelian and the action is given by a cocycle as in 2.5.1.

128

The crossed-products of UHF algebras by product-type act ions studied by B r a t t e l i

in [9] are ap t l y described in terms of groupoids. Let (Xi) be a sequence o f f i n i t e

d isc re te spaces and l e t X =~X i be t h e i r product, wi th the product topology. The

equivalence r e l a t i o n ~ on X, where u ~ v i f f u i = v i fo r a l l but a f i n i t e number of

ind ices, def ines a p r inc ipa l groupoid G. I f the sequence is indexed by N we may de-

f i ne the groupoid G i = { (u ,v) ~ G : uj = vj fo r j ~ i } , w h i c h is elementary. As in

example 1.6, G =UG i is made in to a topo log ica l groupoid which is AF. Since every

po in t o f G O = X has a dense o r b i t , G is minimal.A topo log ica l groupoid isomorphic

to such a groupoid G w i l l be ca l led a Glimm groupoid, because, as we shal l see, i t s

C ~-a lgebra is a UH~or Glimm, algebra.

Let A be an abel ian l o c a l l y compact group. A cocycle c ~ ZI(G,A) w i l l be said

of product type i f i e is o f the form

c(u,v) = ~ c i ( u i , v i ) where c i c ZI(Gi,A)

where G i is the t r a n s i t i v e groupoid on the set X i . We may wr i te

c i ( u i , v i ) = b i ( u i ) - b i ( v i )

wi th b i funct ion from X i in to A. We l e t C i = c i (G i ) = B i - B i where B i = b i ( X i ) . We

may assume that 0 ~ B i . Let us note tha t , by the d e f i n i t i o n o f the topology o f G as

induct ive l i m i t topology, a cocycle of product type is continuous.

1.8.

and l e t c be a cocycle in ZI(G,A) of product type as above.

( i ) The asymptotic range o f c is R (c) = j ~ (i!j_ Ci)"

( i i ) I ts T-set is T(c) ={~ ~ A : V ~ > O, J : I~( i>_j

Proposi t ion : Let G be a Glimm groupoid, A an abe l ian l o c a l l y compact group

( i i i )

ex is ts j such that ~ B i is contained in V. i~ j

( i v ) The asymptotic range o f c at u is R~(c) =

where b i = b i ( u i ) .

B i ) - I f <_ e}.

The cocycle c is a coboundary i f f fo r every neighborhood V of 0 in A, there

f~ join (i!j_ Bi - b i )

o f continuous G-sets, namely, the sets

Proof : The assert ions ( i ) , ( i i ) , and ( i v ) resu l t from the d e f i n i t i o n 1 .4 .3 . The

asser t ion ( i i i ) resu l ts from propos i t ion 1.4.8. We have to check that the hypotheses

o f th i s p ropos i t ion are s a t i s f i e d . The un i t space o f G is compact and G admits a cover

129

• i-ll_T u i s= {~a,ui;,~o~al,u1~j'" " ' ' " " " ~ G : a ~ X ~ , ~ l - [ X~ } 1 J j ~ i J

where i is an in teger and { a b i j e c t i o n of ~c~z X< onto i t s e l f . 1 J

Q.E.D.

B r a t t e l i po ints out in [9 ] tha t the s i m p l i c i t y o f the crossed-product algebras

he considers depends heav i l y on the s t ruc tu re of the group A. This is summarized in

the fo l l ow ing propos i t ion .

1.9. Proposi t ion : Let G be a Glimm groupoid, A an abel ian l o c a l l y compact group and

l e t c be a cocycle in ZI(G,A) of product type.

( i ) I f A is compact and R (c) = A, then G(c) is minimal

( i i ) I f A can be ordered, then G(c) is not minimal.

Proof : The asser t ion ( i ) resu l t s d i r e c t l y from 1.4.16 ( i i ) .

To prove the second asser t ion , we use the no ta t ion given above. We may choose

b i so tha t B i = b i ( X i ) is contained in the pos i t i ve cone P of A and b i is non-decrea-

sing when X i = { 0 , I . . . . . n i } has i t s usual order. Let 0 and 1 denote respec t i ve ly

the sequences 0 = (0,0 . . . . . ) and 1 = ( n l , n 2 . . . . ) in X =4"-FX i . Then the asymptotic range

at 0 o f c R~(c) = ~ ( Z B i ) , is contained in the pos i t i ve cone of A whi le the ' J ~ i ~ j

asymptotic range at i o f c, Rl(c) = .N ( ~ Bi) is contained in the negative cone JqN i~ j

o f A, -P. By 1.4.14 ( i ) , f o r every a e A, the points (O,a) and (1,a) do not have a

dense o r b i t . Therefore, G(c) is not minimal.

Q.E.D.

1.10. Example : The gauge automorphism group of the CAR algebra.

Let us f i r s t def ine the CAR groupoid (CAR stands fo r canonical anticommutation

r e l a t i o n s ) . I t is a Glimm groupoid isomorphic to the groupoid o f the Is ing model.

The pos i t ions of a system of fermions are labeled by a countable set o f ind ices ,

say N- The u n i t space of the groupoid is X = i~T~Xi WnereXi={O,1}.A con f i gu ra t i on u=(u i )

in X t e l l s i f there is a fermion at the place i .As before, G is the p r i nc ipa l groupoid

given by the equivalence r e l a t i o n ~, where two con f igu ra t ions are equ iva len t i f f

they d i f f e r at at most a f i n i t e number o f places. We shal l see tha t i t s C* -a lgeb ra

130

is the C*-a lgebra of the canonical anticommutation re lat ions (see [8] or [29] page

269).

The gauge automorphism group is defined by the product cocycle c c ZI(G,Z),

cal led the "number" cocycle, given by

c(u,v) = ~ u i - v i .

The number cocycle counts the number of par t ic les by which the configurat ions u and

v d i f f e r . With above notat ions, B i = {0,1} and R (c) = Z .

Let us define next the GICAR groupoid (GI stands for gauge invar ian t ) . I t is the

subgroupoid c - l ( o ) . In other words, i t corresponds to the equivalence re la t ion m,

where two configurat ions u and v are equivalent i f f they d i f f e r at at most a f i n i t e

number of places and have the same number of par t ic les (in the sense that c(u,v) = 0).

I ts C*-a lgebra is the subalgebra of f ixed points of the gauge automorphism group ;

i t is cal led the GICAR algebra. I t results from 1.4.17 that the GICAR groupoid is

i r reduc ib le . More information about i t w i l l be given a f te r we introduce the dimension

group of an AF-groupoid.

F ina l ly , l e t us consider the skew-product groupoid G(c). By 1.7 i t is an

AF-groupoid. I t is i r reducib le (by 1.4.13) but not minimal (by 1.9).

The remainder of th is section is devoted exclusively to topological groupoids

which admit a base of open sets consisting of compact open G-sets. Af ter a few

de f in i t i ons , we shall study the example of AF-groupoids.

Let G be a topological groupoid which admits a base of compact open G-sets.

I ts ample semi-group ~ has been defined (1.2.10) as the inverse semi-group of i t s

compact open G-sets. The idempotent elements of ~ are compact open subsets of the

uni t space G O of G. They form a generalized Boolean algebra ~0 (that is, a Boolean

algebra without the assumption that a greatest element ex is ts ) . We define the fo l lowing

equivalence re la t ion on ~0. We shall declare e and f equivalent, and wri te e f ,

i f f there exists s ~ (~ such that e = r(s) and f = d(s), where r(s) = ss -1 and

d(s) : s- ls .

Using terminology common to the theory of von Neumann algebras, one can make the

fo l lowing de f i n i t i on .

131

1.11. D e f i n i t i o n : Let G be a topo log ica l groupoid which has a base o f compact

open G-sets and l e t ~ be i t s ample semi-group. We say that an idempotent element

e of ~ i s f i n i t e i f fo r any idempotent element f , the r e l a t i on e ~ f < e impl ies f = e.

We say tha t G is o f f i n i t e type i f every idempotent element of ~ is f i n i t e and of

i n f i n i t e type otherwise.

We may def ine on ~0 the r e l a t i o n e < f i f f there ex is ts e I and f l such tha t

f . We may also def ine a pa r t i a l add i t ion in ~0 , where two idemDotent e e 1

elements e and f can be added i f f they are d i s j o i n t and e + f is the union of e and f .

We denote by D(G) the o f equivalence classes ~0/~ and by D the quot ien t map of 9 0

onto D(G). We provide D(G) w i th the r e l a t i on D(e) ~ D(f) i f f e < f and wi th a pa r t i a l

add i t i on , where two classes D(e) and D(f) can be added i f f they contain d i s j o i n t

elements e I and f l and then D(e) + D(f ) = D(e I + f l ) . I f G is o f f i n i t e type, the

r e l a t i o n < is an order r e l a t i o n .

1.12. D e f i n i t i o n : Let G be a topo log ica l groupoid which admits a base of compact

open G-sets and l e t ~ be i t s ample semi-group. Assume tha t G is o f f i n i t e type. Then,

i t s dimension range is the set D(G) =(~0/~ w i th the order s t ruc tu re and the pa r t i a l

add i t i ve s t ruc tu re def ined as above.

I t can be shown (c f . [2~ and [27] in the AF case) tha t the dimension range D(G)

of G can be embedded in a unique fashion as a generat ing upward d i rec ted hered i ta ry

subset o f a d i rec ted ordered abel ian group, ca l led the dimension group of G and

denoted by Ko(G ).

The property of being of f i n i t e type is preserved under f i n i t e products, d i s j o i n t

unions and induc t i ve l i m i t s . I t can be shown tha t

Ko(G 1 x G2) = Ko(G1) ® Ko(G2) ,

w i th pos i t i ve cone generated by

K;(G I) K (G 2) ; n

D(G 1 x G2) = {~ m i x n i ; n n n

, m i ~ D(G1), n i ~ D(G2), ~ m i ~ D(G1), ~ n i ~ D(G2)}

132

KO( e Gn) = ~Ko(Gn) ,

where • G n is the d i s j o i n t union of a sequence (Gn) and ~Ko(Gn) is the d i r ec t sum

of the ordered abel ian groups Ko(Gn) ;

k D( • Gn) = {~ m i : k ~ , m i c D(Gi) } ,

m o(ll_~ Gn) = ~ mo(Gn) ,

where l im G is the induc t i ve l i m i t o f an increas ing sequence (Gn) and l im Ko(Gn) is

the induc t i ve l i m i t o f the ordered abel ian groups Ko(Gn) ; and

D(li_~m Gn) = u D(Gn) .

An ordered abel ian group w i l l be ca l led an E l l i o t t group (c f . [25] , [27] and [28]]

i f i t is the induc t i ve l i m i t o f a sequence of ordered groups, each isomorphic to the

d i r e c t sum of f i n i t e l y many copies of T w i t h i t s usual order. The property of being

an E l l i o t t group is preserved under f i n i t e tensor products, countable d i r ec t sums and

countable induc t i ve l i m i t s .

For example, the dimension group of a second countable t o t a l l y disconnected

l o c a l l y compact space X is an E l l i o t t group. Indeed, the dimension range of X is the

(general ized) Boolean algebra of i t s compact open subsets, which may be w r i t t en as an

increasing union of a sequence of f i n i t e Boolean algebras ~ n and i t s dimension

group is the group of cont inuous funct ions wi th compact support of X in to Z

Cc(X,Z ) = { f e Cc(X,Z ) : suppf e~n } , w i th i t s usual order. The dimension range

of the t r a n s i t i v e groupoid on a set o f n elements, where n = 1,2 . . . . ,=, is { 0 , 1 , . . . , n }

and i t s dimension group, which is Z , is an E l l i o t t group. Therefore, the dimension

group o f an AF groupoid is an E l l i o t t group.

The importance of the dimension ranges and of the dimension groups in the

study of AF groupoids is given by the fo l l ow ing p ropos i t ion . The f i r s t asser t ion is

e s s e n t i a l l y p ropos i t ion 3.3 of ~2 ] and the second asser t ion is theorem 3.5 of [52].

Let us note tha t t h i s theorem has a long h i s t o r y in the context o f C* -a lgeb ras

( ~ 5 ] , [18], ~ ] and ~ 7 ] ) .

1.13. Proposi t ion {W.Krieger) :

( i ) The dimension group of an AF-groupoid is an E l l i o t group and every E l l i o t t

133

group occurs as the dimension group of an AF groupoid.

( i i ) Two AF-groupoids are isomorphic i f f the i r dimension ranges are isomorphic.

The AF-groupoids considered in [52] have a compact uni t space, but, as pointed

out there, this assumption can be removed. Given an AF-groupoid G with compact uni t

space, the subgroup of i ts ample semi-group consisting of those G-maps which are

everywhere defined is an ample CLF group in the sense of [52]. Conversely, given an

ample CLF group acting on the space X, the groupoid of the corresponding equivalence

re la t ion on X is AF.

Let us describe the dimension range of the AF-groupoids that we have met in this

section. The dimension group of the Glimm groupoid of the Ising model (and of the

canonical anticommutation re lat ions) is the group Q(2 ~) of rat ional numbers whose deno-

minator is a power of 2, with the order inher i ted f rom~. I ts dimension range is the

segment [0,17 . With the notations of 1.6, the dimension of a cyl inder set C(Zn)

obtained by f i x i ng the spins inside a f i n i t e subset Z n of the l a t t i c e is

D(C(Zn) ) = 2 _iZnI.Therel exists a unique probab i l i t y measure ~ on { - i ,+1} z which

extends D. I t is the unique ergodic invar iant p robab i l i t y measure of the groupoid.

We give in the appendix a computation of the dimension group of the GICAR grou-

poid. I t is the group Z/ I t ] of polynomials in one var iable with integer coef f i c ien ts ,

where the order is given by f > 0 i f f f ( t ) > 0 for every t ~ ] 0 , 1 [ . This is an example

of a Riesz group (cf. [25]). There are uncountably many invar iant ergodic p robab i l i t y

measures, indexed by t c ]0,1land obtained by composing the dimension map with the

point evaluation at t . The measure corresponding to t = ~ is the unique invar iant 2

probab i l i t y measure for the CAR groupoid.

The dimension group of the skew-product of the CAR groupoid and the number

cocycle can be computed in the same fashion as the dimension group of the GICAR

groupoid. I t is the group ]~ ( t ) of rat ional functions with integer coef f ic ients and

whose only possible poles are at 0 and 1, where the order is given by f > 0 i f f

f ( t ) > 0 for every t ~ ] 0 , 1 [ .

Let us look at the re la t ionship between AF-groupoids and AF C * -a lgebras. I t is

due to Krieger ( [52], theorem 4.1) and re l ies essent ia l ly on a resul t of S t r~ t i l~

134

and Voiculescu ( [69 ] , sect ion I of chapter I ) . We give a se l f -conta ined proof which

is e s s e n t i a l l y the same as t he i r s . Let us reca l l that an AF C*-algebra is the induc-

t i v e l i m i t o f a sequence o f f i n i t e -d imens iona l C* -a lgeb ras . Basic references fo r

AF C~-a lgebras are [ 3 ~ , [ i ~ and [8] .

The crux of the proof is the fo l l ow ing lemma about f i n i te -d imens iona l C* -a l geb ras .

1.14. Lemma : Let A be a f i n i te -d imens iona l * - a l g e b r a and A 1 a s u b * - a l g e b r a . Then,

fo r any Cartan subalgebra B 1 of A 1, there ex is ts a Cartan subalgebra B of A which con-

ta ins B 1 and whose normal izer ~ ( B ) , that i s , the inverse semi-group of p a r t i a l

isometr ies a o f A such tha t d ( a ) , r ( a ) ~ B and a(Bd(a))a = Br (a) , contains the norma-

l i z e r ~N~(B1) of B 1 in A I .

Proof : Since A 1 is a sum of simple , -a lgebras , we may assume that A 1 i t s e l f is

simple. The normal izer ~ ( B I ) o f B 1 in A I contains matr ix un i ts ( e i j ) i , j = I . . . . . m

which span A 1, The p ro jec t ion e l l of B 1 decomposes in A in to minimal pro jec t ions :

e11 = f l + . . . + f n . The fami ly ( e i l f j e l i ) i = 1 . . . . . m and j = 1 . . . . . n consists of

orthogonal pro ject ions and is contained in a Cartan subalgebra B o f A. The algebra B 1,

which is spanned by the pro ject ions ( e i i ) i = 1 . . . . . m, is a subalgebra o f B. The

matr ix un i ts ( e i j ) normalize B. Therefore Okrl(B1) is contained i n ~ ( B )

Q.E.D.

1.15. Proposi t ion : Let A be a C * - a l g e b r a . The fo l l ow ing proper t ies are equ iva len t .

( i ) The C* -a lgeb ra A is AF.

( i i ) The C* -a l geb ra A is the C * -a l geb ra o f an AF-groupoid G. Moreover, under

these cond i t ions , the AF-groupoid G is unique up to isomorphism and i t s dimension

range is the dimension range of A (c f . ~ 7 ] ) .

Proof : Suppose that A is an AF C* -a l geb ra and choose an increasing sequence o f

f i n i t e -d imens iona l C* -a lgeb ras A n which def ines A. Construct by induct ion a sequence

o f Cartan subalgebras B n of A n such that Bn+ 1 contains B n and i t s normal izer J~rn+lin~+1

contains the normal izer J~r n o f B n in A n . Let B be the closure o f the union o f the

I B n s. Since~N~ normalizes B m fo r m ~ n, i t normalizes B, hence the ample inverse semi-

group ~n o f B n acts on B. We r e a l i z e B as C * ( X ) , where X is a t o t a l l y disconnected

135

l o c a l l y compact space and we l e t ~=U~n, viewed as an inverse semi-group of pa r t i a l

homeomorphisms of X.The corresponding equivalence r e l a t i on on X y ie lds a p r inc ipa l

groupoid G which is AF because i t is of the form G = u Gn, where G n is the p r inc ipa l

groupoid of the equivalence re l a t i on corresponding to ~n" I t is almost obvious tha t

G n is an elementary groupoid. For, ~n p a r t i t i o n s the atoms of the Boolean algebra Bn

of pro jec t ions of B n in to equivalence classes. Let {YI , . . . . ,Ym} be one of these classes

and l e t Y = Y lV . . . vY m. Then the reduct ion of G n to Y is isomorphic to Y1 x Im, where

I m is the t r a n s i t i v e groupoid on m elements. The lemma al lows the const ruc t ion of

cons is ten t systems of matr ix un i ts in each algebra A n . In other words, there ex is ts

a sect ion k fo r the canonical map of J~ r= uo~C n onto ~. Let C * ( ~n ) be the ( f i n i t e -

dimensional) sub C*-algebra of C*(G) generated by {Xs: s ~ ~n }. There ex is ts an

isomorphism #n of C * ( ~ n ) in to A n such that #n(XS) = k(S) fo r S ~ ~n" Since the

r e s t r i c t i o n of #n+l to C* (~n ) is 0 n, there ex is ts an isomorphism # of W C* (~n)

onto ~ A n whose r e s t r i c t i o n to C* ( ~n ) is O n . I t is isometr ic w i th respect to the

C*-norms of C * (G) and of A, because f i n i t e -d imens iona l * -a lgebras have a unique

C * -norm. Therefore, i t extends to an isomorphism of C*(G) onto A.

The above argument also shows tha t the C * -a l geb ra of an AF-groupoid is AF.

Let us keep the same notat ions as above. The dimension range D(~n ) =~ /~n is

also the dimension range D(An) o f the * -a lgeb ra A n . The dimension range of G, which

is the induc t i ve l i m i t of the dimension ranges D( ~n ), is equal to the dimension

range of the l o c a l l y f i n i t e . -a lgebra uA n. I t is known (e.g. ~ 7 ] , remark 4.4,

page 34) tha t t h i s is also the dimension range of A. Therefore, the uniqueness of the

AF-groupoid G resu l t s from 1.13 ( i i ) .

Q.E.D.

1.16. Coro l la ry : Suppose tha t a C* -a lgeb ra A has two Cartan subalgebras B 1 and B 2

which are both AF and which have countable l o c a l l y f i n i t e ample semi-groups, then

B 1 and B 2 are conjugate by an automorphism of A,

Proof : The groupoids G I and G 2 obtained by 2.4.15 are AF. (Therefore, the 2-cocycles

~i and ~2 are equal to 1). By the previous p ropos i t ion , A is AF and G 1 and G 2 have

the same dimension range. Therefore, they are isomorphic and an isomorphism of G I

136

onto G 2 implements an automorphism of A ca r ry ing B I onto B 2.

Q.E.D.

This is the only r e s u l t we have about the ex is tence and the uniqueness o f Cartan

subalgebras. I t is not known, even in the case of an AF C * - a l g e b r a , whether a

C ~ -a l geb ra may have non-conjugate Caftan subalgebras.

The f o l l o w i n g example shows tha t the d e f i n i t i o n we give o f a Cartan subalgebra

cannot be weakened i f we expect uniqueness.

Let K be the a lgeb ra i c c losure o f a f i n i t e f i e l d , w i th the d i sc re te topo logy .

The m u l t i p l i c a t i v e group o f K is denoted K ~ , i t s a d d i t i v e group is denoted K + and

the dual group of K + is denoted K+. Since K is an increas ing sequence of f i n i t e f i e l d s

K n, K + is the induc t i ve l i m i t o f f i n i t e groups K +n and K+ is the p r o j e c t i v e l i m i t o f

f i n i t e groups K+ As a topo log ica l space K+ is homeomorphic to the Cantor space. n"

The "ax + b" group over K is the sem i -d i r ec t product G = K + x K*, where K*

acts on K + by m u l t i p l i c a t i o n . I t is equipped wi th the product topo logy . We view K + as

a normal abe l ian subgroup of G. Since G has the d i sc re te topo logy , the C * - a l g e b r a

B = C * (K +) is a subalgebra of A = C ~ (G).

1.17. Propos i t ion : Let A and B be as above.

( i ) The C ~ -a lgebra A is AF.

( i i ) The subalgebra B is maximal abe l ian , r egu la r , is the image o f a unique

( f a i t h f u l ) cond i t i ona l expec ta t ion but i t s ample semi-group does not ac t r e l a t i v e l y

f r e e l y on the spectrum K+ of B, hence i t f a i l s to be a Cartan subalgebra.

Proof : ( c f . Dixmier [ 17 ] ) .

( i ) As above, we w r i t e K as union o f an increas ing sequence of f i n i t e f i e l d s K n-

The "ax + b" group over Kn, Gn, is a subgroup of G and G is the union of the Gn'S.

As in 1.15, we see tha t C * ( G ) is the induc t i ve l i m i t o f the C*(Gn)'S, which are

f i n i t e - d i m e n s i o n a l .

( i i ) As an increas ing union of f i n i t e groups, G is amenable. We apply 2 .4 .2 , to

view the elements of C*(G) as func t ions on G vanishing at i n f i n i t y . The elements of

I37

C * ( K +) are those func t ions which vanish outs ide K +. To show tha t B is maximal

abe l ian, we pick an element f o f i t s commutant in A. I t s a t i s f i e s ~b l . f ~e_b I = f fo r

every b I a K +, where Cbl is the po in t mass at b I . E x p l i c i t l y , t h i s gives

f ( a , ( 1 -a )b I +b) = f (a ,b ) fo r every b I ~ K +, a E K*, b ~ K +. Since f vanishes at i n f i -

n i t y , t h i s is on ly possib le i f f ( a ,b ) = 0 when a # 1, tha t i s , f ~ B.

Since K + is a normal subgroup, the normal izer of B contains the elements ex'

where x c G. Therefore B is regu la r .

Let P be a cond i t iona l expectat ion onto B. From the re l a t i ons

( l , b l ) (a,b) ( l , b l ) - 1 = (a , (1-a)b I +b) and

(1 ,b l ) (a,b) = (a,b + bm)

fo r every a s K* and every b,b I ~ K +, we obta in tha t P(a(a,b)) =

P (E (a , ( l - a )b l +b) = C ( l - a )b l * P(a(a,b) )" Thus, i f a # 1, P(a(a,b)) is i n v a r i a n t under

t r a n s l a t i o n . Since i t vanishes at i n f i n i t y , i t must be zero. This shows tha t the

r e s t r i c t i o n of P to Cc(G ) is the r e s t r i c t i o n map of Cc(G ) onto Cc(K+). On the other

hand, i t resu l t s from 2.2.9 tha t th i s r e s t r i c t i o n map is pos i t i ve and bounded. Hence

i t extends un iquely to a cond i t iona l expectat ion of C* (G) onto C* (K+ ) , which is

s t i l l given by r e s t r i c t i n g a func t ion to K +. I t is c l e a r l y f a i t h f u l .

To show tha t the ample semi-group of B does not act r e l a t i v e l y f r e e l y on K +,

we note tha t the element C(a,b) o f the normal izer o f B induces the homeomorphism s a

of i t s spectrum K +, where Sa(× ) = ax and a×(b) = x(ab) fo r × ~ K +. I f a # I , the

set o f f i xed points o f s a is reduced to the i d e n t i t y character 1, hence is not open

in K + .

Q. E.D.

We have not been able to determine whether the exact sequence

"~+~(B) ÷~(B) # ~(B) +%

s p l i t s or not.

1.18. Remark : The C* -a lgeb ra A is the C* -a lgeb ra of the t ransformat ion group

(K+,K*) where the act ion of K* on K+ is described above. Since Y = K+\ { I } is an

i nva r i an t open subset o f K+, A is an extension of C * (Y x K*) by C ~(K*) (2 .4 .4 ) .

One can show that the dimension range of the groupoid Y × K* is the segment [O,p[

138

of the dimension group Q(p®) of ra t i ona l numbers whose denominator is a power of p,

where p is the cha rac te r i s t i c of the f i e l d K. Therefore, C* (Y × K*) is a matroid

algebra wi thout un i t of type Mp~ p (nota t ion of [18]) . On the other hand, the

C* -a l geb ra C*(K*) is the C* -a lgeb ra of the Cantor space. I t resu l ts from [25],

sect ion 5.1, that the dimension group of A is an extension of Q(p~) by the dimension

group o f the Cantor space.

Let us mention here, w i thout g iv ing the d e t a i l s , that such extensions are

character ized, up to equivalence, by measures on the Cantor space. E x p l i c i t l y , one

f inds tha t Ko(A ) = Q(p~) x C(K* ,Z ) and that an element (q , f ) is pos i t i ve i f and

only i f q + #( f ) is pos i t i ve where # is the measure on K*, constructed as fo l lows .

Let (n i ) be a sequence o f integers such that n i d iv ides ni+ 1 and n I = I , l e t

ni qi - 1 qi = p and l e t f i - fo r i > 2 and f l = p" Real ize the space K ' a s the

qi -1-1

product space ~ {1,2 . . . . . f i }. The measure # is concentrated on the points (a i ) i=1

with a i = 1 fo r i large enough. I f k is the l as t index i fo r which a i # 1 (or , i f

a I = i fo r every i set k = I) the measure of the po in t (a i ) is P i i _ i j .

• , , ! ~ q j + l l j=k qj -1 L J

The dimension range of A is the segment [O,c], where c is the element (p-1,1) of

Q(p~) × c(~*, ~ .

Another method to check that the subalgebra B is not a Cartan subalgebra is to

determine i t s dimension range and i t s dimension group r e l a t i v e to A. I ts dimension

group is an extension of Q(p~) by Z .

2. The Groupoids O n

The aim of th is sect ion is to e x h i b i t the C* -a lgebras generated by isometr ies

introduced by J.Cuntz in [ l ~ a s the C* -a lgeb ras o f a groupoid. The groupoids we

construct are not p r inc ipa l and we do not know i f these algebras can be rea l i zed as

the C* -a lgeb ras o f a p r inc ipa l groupoid. Nevertheless, th is descr ip t ion o f the Cuntz

algebras reveals much of t h e i r s t ruc ture . I t a lso makes apparent the r e l a t i onsh ip

between these algebras and some inverse semi-groups.

139

We s t a r t w i th a crossed product cons t ruc t ion prompted by the represen ta t ion o f

the Cuntz algebras as a crossed product (sec t ion 2 of [ 15 ] ) . We inc lude the case

n = I , which w i l l g ive the a lgebra o f the b i c y c l i c semi-group, s tud ied by Barnes in

[1 ] .

For every n = 1,2 . . . . . ~, we def ine the f o l l o w i n g AF-groupoids G n. The groupoid

G 1 is the compact space Z = Z u { ~ } , one-po in t compac t i f i ca t i on o f the space o f in tegers

wi th i t s d i sc re te tooo logy . We reca l l tha t i t corresponds to the equivalence re-

l a t i o n u ~ v i f f u = v on ~ .

For n l a rge r than 1.but f i n i t e , the groupoid G n corresponds to the equivalence

r e l a t i o n u ~ v i f f u i = v i fo r a l l but f i n i t e l y many i ' s on the compact space

{0,1 . . . . . n - l } w i th the product topo logy . This is a Glimm groupoid. I t s dimension

group is the group ~(n ~) o f r a t i o n a l numbers whose denominator is a power of n, w i th

the order i nhe r i t ed from ~ and i t s dimension range is the segment [0,1] ~. _ z

The un i t space o f the groupoid G is the space G 0~ = {u e ~ : u i = 0 f o r i

s u f f i c i e n t l y small and uj = ~ fo r every j > i i f u i = ~ } , where ~-~= N u{~} . The

cy l i nde r sets Z(~) , where ~ = ( . . . . O, jk . . . . . jk+L) w i th k e ~ , L e~q and Jk+i c ~ ,

and t h e i r complements form a subbase of open sets f o r a topology on G~. This topo logy

is l o c a l l y compact and t o t a l l y d isconnected. The cy l i nde r sets Z(m are compact. We

de f ine , fo r u ~ G~, k(u) as the smal les t index i such tha t u i ~, i f i t ex i s t s and

as ~ i f u i < ~ f o r every i . The groupoid G corresponds to the equlvalence r e l a t i o n

on G~ : u ~ v i f f k(u) = k(v) and u i = v i fo r a l l but f i n i t e l y many i ' s . One checks

as in the example of the Glimm groupoids tha t i t is an AF-groupoid. The c losure of the

o r b i t o f a po in t u is [u] = {v : k(v) ~ k (u ) } . In p a r t i c u l a r , there are dense o r b i t s .

The i n v a r i a n t open sets form a decreasing sequence (U i ) , i m~, where U i =

{u : k(u) > i } . The dimension group of the groupoid G is the l ex i cog raph i ca l d i r e c t

sum i B y ( c f . 5.3 o f [28] ) and i t s dimension range is the whole p o s i t i v e cone. Fur ther

references to the AF C * - a l g e b r a s whose dimension group is t o t a l l y ordered can be

found in [28].

In each case, there ex i s t s a natura l s h i f t #0 on the u n i t space of G n which

normal izes the ample semi-group of Gn, tha t i s , such tha t f o r any G-map s in the ample

semi-group of G n, #0 o s o #0-1 is a lso in the ample semi-group o f G n. E x p l i c i t l y ,

140

f o r n = i , the s h i f t ~0 one_ sends u i n t o u - 1 i f u i s f i n i t e and ~ i n t o ~. For

n : 2, ~0 i s g iven by ¢Ou = v where v i = u i_ 1. The s h i f t ¢0 induces an automorphism

o f the l o c a l l y compact g roupo id G n. E x p l i c i t l y , # i s g i ven by ~ ( u , v ) = ( ~ O ( u ) , ~ O ( v ) ) .

We l e t Z ac t on G n by z + ~z and form the s e m i - d i r e c t p roduc t G n x ~ ( s e e 1 .1 .7 and

the beg inn ing o f the p r o o f o f 2 . 3 . 9 ) . I t i s an r - d i s c r e t e g roupo id a d m i t t i n g the coun-

t i n g measures on the f i b e r s as Haar system.

F i n a l l y , we d e f i n e f o r each n the f o l l o w i n g subset o f u n i t space o f G n. For

n = I , O~ = ~ = I N u { ~ } . For 2 _< n < ~, O On = {u ~ { 0 , 1 , . . . , n - I } ~ : u i = 0 f o r i < 0 } .

w i l l be i d e n t i f i e d w i t h {0 ,1 . . . . . n - l } ~ For n = ~, 00 = {u s G O : u i = 0 f o r i < 0 ) . I t

I t w i l l be i d e n t i f i e d w i t h {u s ~ : u i = ~ ~ u j = ~ f o r e v e r y j ~ i } . Each o f

these subsets 00 i s c losed in G O hence is a compact space n n ' "

2 .1 . D e f i n i t i o n : Le t n = 1,2 . . . . . ~. The Cuntz g roupo id On is the r e d u c t i o n o f the

0 s e m i - d i r e c t p roduc t G n × # ~ t o the c l osed subset O n o f i t s u n i t space ( i d e n t i f i e d w i t h

the u n i t space o f Gn).

Le t us spe l l ou t the a l g e b r a i c s t r u c t u r e o f the g roupo ids O n . For n = 1,

01 = { ( u , z ) ~ ~ - ] x Z : u + z ~ } , where ~ + z . . . . The m u l t i p l i c a t i o n i s g i ven by

( u , z ) ( u + z , z ' ) = ( u , z + z ' ) and the i n v e r s e o f ( u , z ) is (u + z , - z ) . The range u n i t

o f ( u , z ) i s u and i t s domain u n i t i s u + z. For n g r e a t e r than 1 but f i n i t e , N

O n = ~ ( u , v , z ) e { 0 , 1 . . . . . n - l } × {0 ,1 . . . . . n - l } × ~ :

u i = v i _ z f o r a l l but f i n i t e l y many i ' s ) .

The m u l t i p l i c a t i o n i s g i ven by ( u , v , z ) ( v , w , z ' ) = ( u ,w , z + z ' ) a n d the i n v e r s e o f ( u , v , z )

i s ( v , u , - z ) . The range u n i t o f ( u , v , z ) i s u and i t s domain u n i t i s v . In the case

0 = { ( u , v , z ) e 00 × 00 × Z : u i = v i _ z f o r a l l but f i n i t e l y many i ' s and

k (u ) = k (v ) = z in the case when k(u) o r k ( v ) i s f i n i t e } .

The m u l t i p l i c a t i o n and the i n v e r s e are g i ven as above.

The nex t task i s to de te rm ine the ample semi -g roup o f the g roupo ids O n . Le t us

f i r s t d e f i n e the Cuntz i n v e r s e semi -g roup O n , i n t r o d u c e d i m p l i c i t l y i n the f i r s t

s e c t i o n o f [ 1 ~ . The semi -g roup 01 i s the b i c y c l i c semi -g roup [ 11 ] .

141

2 .2 . D e f i n i t i o n : Le t n = 1,2 . . . . . ~. The Cuntz i n v e r s e semi -g roup O n i s the

semi -g roup c o n s i s t i n g o f an i d e n t i t y 1, a zero e lement 0 and a l l the words in the

l e t t e r s P i ' q i w i t h i = I . . . . . n, s u b j e c t t o the r e l a t i o n s q j P i = 0 i f i # j and

q iP i = 1.

Le t us r e c a l l the n o t a t i o n s o f [ 1 5 ] . Le t n = 1,2 . . . . . ~. Given k c ~ ,

n n l e t W kn = {~ = ( J l ' " " J k ) : J i ~ {0 ,1 . . . . . n - l } } , W 0 = {0 } and W n~ = k=OU W k-

~ = . and q~ . . q j Then, For = ( J l . . . . . Jk ) W~, l e t p~ PJIPJ2 " 'PJk = qJkqJk-1" 1"

i t i s shown in [15] (lemma 1.3) t h a t any word in p i q i may be u n i q u e l y w r i t t e n pmq~

W n . (p~qB) - I w i t h ~,~ e ~ The semi -g roup O n i s an i n v e r s e semi -g roup w i t h = pBqm, 1-1=1

0 = W~} { 0 , 1 } . The and 0 "1 = O. I t s se t o f i dempo ten t e lements i s O n {pmqm :m

0 o r d e r on O n i s ( i I . . . . . i k) 2 ( J l . . . . . j ~ ) i f f k Z Land i m = Jm f o r m = 1 . . . . . L.

The compact open G-sets o f G n x c Z a re o f the form S × {z } where S i s a

compact open G-set o f G n and z m Z . T h e r e f o r e , the compact open G-sets o f O n are o f

the form { ( u , z ) : u e S} where z c Z and S is a compact open subset o f ~4 such t h a t

S + z c N in the case n = 1 and o f the form { ( u , v , z ) E 00 x 00 x Z : ( u , v ( - z ) ) e S} n n

where S is a compact open G-set o f G n and [ v ( - z ) ] i = v i _ z , in the case n > 2. In

. . . W n p a r t i c u l a r , l e t us d e f i n e , f o r eve ry n = 1 ,2 , ,~ and e v e r y m,B ~ ~ the f o l l o w i n g

compact open G-sets o f O n . For n = 1, S(m,B) = { ( u , ~ ( ~ ) - L(m)) : u e [ ~ ( ~ ) , ~ ] } ,

where the l e n g t h L(m) o f m is k i f m is in W~. For n > 2, S(m,~) =

{ ( ( ~ , u ) , ( B , u ) , ~ ( ~ ) - ~ ( B ) ) : u ~ o ~ ) .

2 .3 . P r o p o s i t i o n : Le t n = 1,2 . . . . . =. The map which sends P~qB i n t o S ( ~ , ~ ) , 0 i n t o

0 i s an isomorph ism o f the i n v e r s e semi -g roup (O n i n t o the ample semi -g roup and 1 i n t o O n

o f the g roupo id O n . I t s image, which w i l l a l so be denoted On, genera tes the ample

semi -group in the sense t h a t

( i ) eve ry compact open se t o f 00 may be w r i t t e n as the d i f f e r e n c e A \ B o f n

two sets A and B which are both a f i n i t e d i s j o i n t un ion o f e lements o f 0 0 . n

( i i ) e v e r y compact open G-set o f O n may be w r i t t e n as a f i n i t e un ion u E iS iF i 1

where (E i ) and ( F i ) a re two f a m i l i e s o f d i s j o i n t compact open sets i n O0 n and the

S i ' s are in O n .

142

Proof : The map is c l e a r l y one- to-one. In order to show tha t i t is a homomorphism,

i t su f f i ces to consider the generators Pi and q i ' s . Let us de f ine , f o r m and B in

W~,P = S(~,O) and QB = S(O,B). Thus, S(~,F) = P Q~. We w r i t e Pi ins tead o f P ( i ) .

Then, the f o l l ow ing r e l a t i o n s are s a t i s f i e d : QjPi = @ i f i # j , OiP i = 0 0 PiPj=P( • n ' i , j )

and QiQ j = Q ( j , i ) " Therefore, the map is an isomorphism of O n in to the ample semi-

group o f O n .

The image of the idempotent element pmqm is the i n te rva l [~(m),~] in the case

n = 1 and the cy l i nde r set Z(~) o f 0 0 in the case n > 2. In the case n - I , the n

asser t ion ( i ) is c l e a r . I t is a lso c lea r in the case 2 E n < ~ since every compact

open set o f { 0 , I . . . . . n - l } m is a f i n i t e d i s j o i n t union o f c y l i n d e r sets Z(~). In

the case n = =, i t su f f i ces to check tha t the sets A\.B, where both A and B are unions

o f c y l i n d e r sets Z(m) form a base fo r the topology o f 00. This is immediate from the

d e f i n i t i o n o f the topology of 00.

The l a s t asser t ion is a lso c l e a r . For example, in the case n > 2, the G-set

{ (u ,v ,O) : (u ,v ) ~ S} where S corresponds to the t r a n s p o s i t i o n (~,u) ÷ (~ ,u ) , w i th

m,B e W n belongs to 0 n. k '

0 E.D.

2.4. Remark : The groupoid O n has the p roper ty o f having i t s ample semi-group

generated by the inverse semi-group O n . Two quest ions a r i se fo r which we have no

answer. Given an inverse semi -g roup~ , does there e x i s t an r - d i s c r e t e groupoid G

whose ample semi-group o f compact open G-sets is generated by ~ and covers G ?

What kind o f uniqueness can we expect ?

Let us note tha t the r e a l i z a t i o n o f an inverse semi-group ~ as a semi-group

o f G-sets in t roduces an ex t ra s t ruc tu re on ~ and embeds i t s idempotent elements

i n to a Boolean a lgebra and a l lows the d e f i n i t i o n of a p a r t i a l add i t i on : S and T can

be added provided tha t r(S) n r (T) = ~ and d(S) n d(T) = ~, then S + T is the union

of S and T. For example, we have int roduced in the case 2 < n < ~ the r e l a t i o n n

PiQi = I . i=1

2.5. P ropos i t i on :

( i ) For every n = 1,2 . . . . . ~, the groupoid O n is amenable.

143

( i i ) For n = 1, j ~ i s an open i n v a r i a n t set f o r 01 . The reduct ion o f 01 to ~ is

the t r a n s i t i v e groupoid on ~ and the reduct ion of 01 to {~} is the group Z -

( i i i ) For n > 2, the groupoid O n is min imal .

( i v ) For every n = 1,2 . . . . . ~, the groupoid O n has a base o f compact open G-sets

and is o f i n f i n i t e type.

Proof :

( i ) We have const ructed O n as the reduct ion of a sem i -d i r ec t product . L.~e may

apply 2.3.7 and 2 .3 .9 .

( i i ) The open subset o f IN of IN is c l e a r l y i n v a r i a n t . We may def ine the

isomorphism of 011 ~ onto IN×IN which sends (u ,z) in to (u,u + z ) . The i so t ropy group

o f 01 at {~} i s i .

( i i i ) The groupoid O n induces the equivalence r e l a t i o n ~ on i t s u n i t space,

where u ~ v i f f there ex i s t s z ~ s u c h tha t u i = v i_ z f o r a l l but f i n i t e l y many

i ' s . Hence every o r b i t meets every c y l i n d e r set Z(~) , where ~ ~ W n fo r 2 < n < ~,

and every c y l i n d e r set Z ( ~ Z ( ~ j ) , where ~,Bj ~ W~for n = ~. This shows tha t every

o r b i t is dense.

( i v ) The G-sets SE, where S c O n and E is a compact ooen set in OOn c o n s t i t u t e

a base fo r the topology o f O n . Since, in O n , p iq i is equ iva len t to 1, the groupoid

O n is of i n f i n i t e type.

Q.E.D.

Let us reca l l the d e f i n i t i o n o f a represen ta t ion o f an inverse semi-group on a

H i l b e r t space given by B. Barnes in [ 1 ] , page 363.

2.6. D e f i n i t i o n : A rep resen ta t ion of an inverse semi-group ~on a H i l b e r t space H

is an inverse semi-group homomorphism of ~ i n t o an inverse semi-group o f p a r t i a l

i somet r ies o f H.

Let V be a represen ta t ion of the inverse semi-group O n , n = 1,2 . . . . . ~. The

images S i = V(Pi) o f the generators Pi are isomet r ies w i th mutua l l y orthogonal ranges.

Conversely, any sequence (S i ) i = i , . . . . n of i somet r ies w i t h mutua l l y or thogonal ranges

def ines a unique rep resen ta t ion V of O n such tha t V(Pi) = S i f o r every i = 1 . . . . . n.

144

n In the case 2 _< n < ~, we requ i re t h a t ~ SiS i = ] .

i=1

2.7. P ropos i t i on : Let n = 1,2 . . . . . ~. There is a b i j e c t i v e correspondence between n

the r e p r e s e n t a t i o n s V of O n on sepa rab le H i l b e r t spaces , such t h a t ~ V(Pi)V(qi) = 1 1

in the case 2 < n < ~, and the r ep resen ta t i ons o f C* (On) on separable H i l b e r t spaces.

Proof : Suppose t h a t L is a r e p r e s e n t a t i o n o f C *(On) Then, by 2 .1 . 20, i t g ives

by r e s t r i c t i o n a r e p r e s e n t a t i o n o f the ample semi-group o f O n , hence a r e p r e s e n t a t i o n n

o f O n . In the case 2 5 n < ~, the r e l a t i o n ~ E i = I , where E i is the c h a r a c t e r i s - i=1

t i c f unc t i on of the c y l i n d e r set Z ( i ) holds in C * , (On) and g ives the r e l a t i o n n

Z sis = i i=1

Converse ly , suppose\ that V is a r e p r e s e n t a t i o n o f O n such t h a t , in the case n ZI 0 2 ~ n < =, S iS i*= 1, where S i = V(P i ) . I t s r e s t r i c t i o n to the set O n o f idempotent

i= e lements, which w i l l be denoted M, is a monotone p r o j e c t i o n - v a l u e d f u n c t i o n , t ak ing

the va lue 0 a t 0 and the va lue i a t I • I t is f i n i t e l y a d d i t i v e in the case 2 s n < ~, n

because o f the r e l a t i o n i~ 1 S iS* = i = I . We w i l l extend i t to a f i n i t e l y a d d i t i v e pro-

j e c t i o n - v a l u e d measure on the Boolean a lgebra ~ n o f compact open subsets o f 00. In n

the case n = 1, any compact open subset o f ~ i s a f i n i t e d i s j o i n t union o f d i f f e r e n c e

o f elements o f O . Thus, i f A = u B i \C i w i th T i , C i eO and C i c B i , we de f i ne i = l

M(A) = 1~=1= M(B i ) - M(Ci) . Because the order o f ~ is t o t a l , M(A) is we l l de f ined and

M is f i n i t e l y a d d i t i v e . In the case 2 ~ n < ~,any compact open subset o f ~N

{ 0 , i . . . . . n - l } is a f i n i t e d i s j o i n t union of elements o f O 0 Thus, i f A = u B i , n" i=1

0 w i th B i ~ On, we de f i ne M(A) = ~ M(Bi ) . This is wel l de f ined and a d d i t i v e because

0 i = I M is a d d i t i v e on O n . In the case n = ~, we f i r s t extend M to the elements o f 0 which

0 are a f i n i t e d i s j o i n t union o f elements o f O n . Since every element A o f O~ is the

d i f f e r e n c e o f two such e lements, say A = B\C w i th C c B, we may de f i ne M(A) = M(B)-M(C).

One shows as in the case n = 1 t h a t M is we l l de f ined and a d d i t i v e .

Having extended M to the Boolean a lgebra ~ n ' we may extend V to a r e p r e s e n t a t i o n

o f the ample semi-group o f O n . We know from 2 . 3 . ( i i i ) t h a t every compact open G-set

o f O n may be w r i t t e n as a f i n i t e union S = u EiSiF i where (E i ) and (F i ) are two 1

~S f a m i l i e s o f d i s j o i n t elements o f ~ n and the S i are in O n . We de f ine

145

g V(S) = ~ M(Ei)V(Si)M(Fi). I t is a par t ia l isometry and i t does not depend on the

I way S has been wr i t ten. Moreover, i t is an inverse semi-group homomorphism.

The pair (V,M) is a covariant representation of O n (cf.2.1.20) and can be exten-

ded to a representation of C*(On). Exp l i c i t l y , every f c Cc(On) may be wr i t ten

00 f = Z1 hixsi where h i aCc(n) and S i is a compact open G-set of O n . We define L(f) =

M(hi)V(Si). A computation s imi lar to one given in the proof of 2.4.15 shows that I L( f ) is well defined. Moreover, the map L so defined is a representation of Cc(On)

continuous for the inductive l i m i t topology. Since r -d iscrete groupoids with Haar

system have su f f i c i en t l y many non-singular Borel G-sets, we know from 2.1.22 that

L extends to a representation of C* (On).

Q.E,D.

2.8. Remarks :

( i ) In order to study the representations of an inverse semi-group ~on a

Hi lber t space, B. Barnes makes use in [1] and [2] of i t s Banach .-algebra 1 ( ~ ) .

He shows in par t icu lar that LI(~) has a fa i t h fu l representation. The example of O n

suggests another approach. One can t r y f i r s t to real ize the inverse semi-group as a

generating subsemi-group of the ample semi-group of a groupoid G and then define the

C* -algebra o f ~ as C*(G). The example of the b icyc l ic semi-group 01 is studied

in [1] (section 7). The descript ion of i t s i r reducib le representations given there

can also be obtained from 2.5. ( i i ) .

( i i ) The C*-algebra C *(On) is generated by the isometries Pi ' i = i . . . . . n.

Indeed these isometries generate OnaS an inverse semi-group. Moreover O~ generates

the Boolean algebra ~n of compact open subsets of OOn" Therefore, the C*-algebra

generated by the Pi 's contains Cc(On). I t must be C* (On). Thus C*(On) i s , for

n ~ 2, one of the C*-algebras studied by J. Cuntz in [15]. I t is shown there that

such an algebra is simple. We can prove i t d i r ec t l y . Indeed the groupoid O n is ame-

nable, minimal and essent ia l l y pr incipal (de f in i t ion 2.4.3). Hence we may apply

proposition 2.4.6.

We have seen (1.1.7) that the semi-direct product G n ×¢2 has a natural cocycle

146

c n c ZI(Gn × ¢ Z , Z ) , namely the cocycle given by Cn(X,Z ) = z. I ts r e s t r i c t i o n to the

reduct ion G n Z I ( o I , Z ) x¢ ~ iO ~ is s t i l l a cocycle. E x p l i c i t l y , fo r n=l , c I ~ is

def ined by Cl(U,Z ) = z and for n > 2, c n c z l ( O n , ~ ) is def ined by Cn(U,V,Z ) = z.

We may observe that the " f i xed po in t " groupoid cnl(O) bears some resemblance wi th

G n. Indeed, fo r n = 1, c n l ( O ) i s the un i t s p a c e ' o f 01 . For 2 < n < ~, c n l ( O ) i s the

Glimm groupoid given by the equivalence r e l a t i o n u ~ v i f f u i = v i fo r a l l but

f i n i t e l y many i ' s on {0,1 . . . . . n - l } For n = ~, c~1(0) is the AF groupoid given

by the equivalence r e l a t i o n u ~ v i f f k(u) = k(v) and u i = v i fo r a l l but f i n i t e l y

many i ' s on 00. I ts dimension group is the lex icograph ica l d i r e c t sum 77 and i~IN

i t s dimension range is the segment [ 0 , i ] , where 1 = (1,0,0 . . . . ).

The fo l l ow ing resu l t , due to Olesen and Pedersen ( [ 58 ] ) , is i n te res t i ng because

i t exh ib i t s the d i f f e r e n t behavior o f the O n groupoids, in comparison to the AF

groupoids, wi th respect to KMS measures. The d e f i n i t i o n o f (c,6) KMS measures has

been given in 1.3.15.

We w i l l make use o f the r e l a t i o n du ' s - I (u) = D ' l ( us ) , where D is the modular d~

func t ion of u and s is a G-map, establ ished in 1.3.18. ( i i i ) and 1.3.20.

2.9. Proposi t ion : Let n = 1,2 . . . . . ~ and l e t c n c Z1(On,~) be as above. Then

( i ) i f n = 1, there are no (cI,B)-KMS p r o b a b i l i t y measures fo r 6>0 and there

ex is ts a unique (Cl,~)-KMS p r o b a b i l i t y measure fo r ~ < 0 ;

( i i ) i f n > 2, there are no (Cn,~)-KMS p r o b a b i l i t y measures fo r ~ # logn and

there ex is ts a unique (Cn,6)-KMS p r o b a b i l i t y measure fo r ~ = logn.

Proof :

( i ) Since d - l ( u ) = { ( v ,u -v ) : v ~ } , i f u is f i n i t e and

d - l (~ ) = { (~ ,z) : z E Z } , Minlcl), which is the set of un i ts u such that the res-

t r i c t i o n of c I to d - l (u ) is non-negat ive, is empty, whi le the set MaXlCl)Of un i ts u

such that the r e s t r i c t i o n of c I to d - l ( u ) is non pos i t i ve is {0} . Therefore there is

no ~ KMS measure and the po in t mass at 0 is the unique -~ KMS p r o b a b i l i t y measure.

Suppose that ~ is a KMS p r o b a b i l i t y measure on N a t a f i n i t e ~. Let Q be the

G-set { ( u , l ) : u ~ } a n d l e t q be the corresponding G-map. For every compact open

subset A o f ~ , the fo l l ow ing equa l i t i e s hold :

147

u(A-q) = ~×A(U) d ( ~ . q - 1 ) ( u )

= ~XA(U) D - l ( u q ) du(u)

= e ~ ~ ( A ) .

In p a r t i c u l a r , f o r every i E N , ~ { i + i } = e ~ u { i } . Since ~ i s r e q u i r e d to be a

p r o b a b i l i t y measure, t h i s i s poss ib l e f o r B ~ 0 on l y . Then u i s un ique l y de f i ned by e Bi

p { i } - - - i f B < 0 and by ~ {~} = i f o r B = O. 1 - e 8

( i i ) Suppose 2 ~ n < ~. Then M i n ~ n )= Max(Cnl= 0 because d - l ( u ) =

{ ( v , u , z ) : v ~ u and z E 7 } and the re are no KMS measures a t i n f i n i t y . Let ~ be a KMS

p r o b a b i l i t y measure on 0 0 a t a f i n i t e ~ We know (1 .3 .16 ) t h a t u i s cn l (O) i n v a r i a n t •

however the Glimm groupo id cn l (O) has a unique i n v a r i a n t p r o b a b i l i t y measure, because

o f the s t r u c t u r e o f i t s d imension range. This i s the measure ~n de f i ned by

W ~ n(~(m) - ~(B)) un(A ) un(Z(m)) = n -~(m) f o r every m c n" Since ~n(A.pmq~) =

W ~ the modular f u n c t i o n o f ~n f o r every compact open se t A and every p a i r (~,B) in n '

w i t h respec t to 0 ~ n i s Dn(U,V,Z ) = n -z = e x p ( - l o g n c ( u , v , z ) ) Thus ~ must be equal to

log n.

( i i i ) Suppose n = ~. Since d - l ( u ) { ( v , u , k ( v ) - k (u ) ) : v ~ u} i f k(u) i s f i n i t e

and d - l ( u ) = { ( v , u , z ) : v ~ u, z c ~ } i f k(u) i s i n f i n i t e , Max(c I i s a lways empty

and Min(c ) = {~} where = denotes the sequence (~ ,~ , . . . ) . Thus there is no }<MS

measures a t ~ = -~ and the p o i n t mass a t { = } , ~ , i s the on ly KMS p r o b a b i l i t y

measure a t B = ~. There cannot be any KMS p r o b a b i l i t y measures a t a f i n i t e ~ because

~ , which i s the on ly p r o b a b i l i t y measure i n v a r i a n t under c ~ l ( o ) , i s not quas i -

i n v a r i a n t under 0 .

Q.E.D.

2.10. Remark : I f G i s the g roupo id o f a t r a n s f o r m a t i o n group (U,S) , where S i s a

subgroup o f ~ , the on ly p o s s i b l e KMS p r o b a b i l i t y measures f o r the cocyc le c ( u , s ) = s

are i n v a r i a n t p r o b a b i l i t y measures.

Appendix.

THE DIMENSION GROUP OF THE GIDAR ALGEBRA

We have seen tha t AF groupoids are c l a s s i f i e d by t h e i r dimension ranges (3.1.13.

i i ) . Therefore, the computat ion of the dimension range is the essent ia l step in the

study o f AF groupoids. The problem can be s p l i t i n to two pa r t s , f i r s t the computation

o f the dimension group, second the de termina t ion of the dimension range as an upward

d i rec ted he red i t a r y subset of the p o s i t i v e cone o f the dimension group. The f i r s t par t

i s more d i f f i c u l t . An E l l i o t t group, tha t i s , the dimension group of an AF groupoid ,

is usua l l y given as an i nduc t i ve l i m i t . Although some in fo rmat ion can be read o f f

from the corresponding diagram, in p a r t i c u l a r i t s ideal s t r uc tu re (see [ 8 ] ) , and one

can decide when two diagrams g ive isomorphic dimension groups, i t is o f g rea t i n t e -

res t to have an i n t r i n s i c d e f i n i t i o n of the dimension group. This is why t h i s compu-

t a t i o n is included here, which is probably known to o thers .

Let us reca l l the d e f i n i t i o n s : the CAR groupoid is

CAR = { ( u , v ) ~ U x U : u i = v i a . e . } where U = {0 ,1} ~ the GICAR groupoid is the

subgroupoid GICAR = c - l ( o ) where c (u ,v ) = ~u i - v i . I t s ample semi-group cons is ts of J J

the G-maps of CAR which "preserve the number o f p a r t i c l e s " . This means # ( u . s ) i = # u i

f o r j l a rge enough. Wr i t i ng GICAR as an i nduc t i ve l i m i t , we ob ta in the f o l l o w i n g

i nduc t i ve system f o r i t s dimension group :

li il . . . . ~ n , ~ n + l ___, . . . n = 1,2 . . . .

149

P r o p o s i t i o n : The d imension group o f GICAR is ~ [ t ] = { po l ynom ia l s in t

w i t h i n t e g e r c o e f f i c i e n t s } w i t h usual a d d i t i o n and o rde r f > 0 i f f f ( t ) > 0 f o r any

t a ] O , I [ .

Proof :

(a) We f i r s t observe t h a t f o r a g iven n and p = 0 . . . . . n,

(1) t p ~ ( n - p ~- ~ t k t ) n - k : k -D / ( I -

k=O

and

11 L k=0

n t k t ) n - k (b) ~le i n t r oduce e k = (1 - f o r n = 1,2 . . . . and k = 0 . . . . . n

and remark t h a t

n ' s genera te ~ [ t ] , (2) the e k

(3) f o r f i x e d n, the e n' k s are l i n e a r l y independent ,

n n+l n+l (4) e k = e k + ek+ 1 n = 1,2 . . . . and k = 0,1 . . . . . n.

Hence, as a g r o u p ~ [ t ] i s the l i m i t o f the i n d u c t i v e system. Let us determine

the o rde r the system induces o n , I t ] . By d e f i n i t i o n , f > 0 i f f f o r s u f f i c i e n t l y

n t k n-k l a rge n, the c o e f f i c i e n t s ~k o f the expansion f = ~ ~k (1 - t ) are non- k=0

nega t i ve .

(c ) We show t h a t f > O and g > 0 imp ly fg > O. Indeed,

m t )m- k f = ~ >'k t k (1 - ~k -> 0 , k = O . . . . . m,

k=O

n g = ~ ~ t L ( I - t ) n-~ ~ >_ O, ~= O . . . . . n,and

~=0

m+n fg = ~ ( ~ X k ~g) t J (1 - t ) m+n-j w i t h

j =0 k+9~=j

~k ug~ >- 0 . . . . , m+n. k+~ =j

150

(d) Let f E Z [ t ] such that f ( t ) > 0 fo r t ~ [ 0 , I ] , then f > O. We wr i te

m ~ t - ! t - P-- ! f = Z apt p and fn = ~ a p t n . . n n > m.

p=O p=O i - 1 ' i - p-1 ' n n

Since fn converges to f uni formly on [0,1]~ fo r n s u f f i c i e n t l y large,

f n ( t ) > 0 fo r ~-E [0,1] .- Th~n-, . . . . . . . . . . . . .

m m n f = ~ apt p = ~ a o ( I

: ~ ~ ap k- (1 - t ) , and k=O p=O

m n-p x p=O p=O

k ( k - 1 ) . . . ( k - p + l ) = ( ~ ) f n (~) > O. n(n-1) (n-p+1)

(e) We conclude that fo r a non zero f e ~ [ ~ , f > 0 i f f f ( t ) > 0 fo r t 0 0 , 1 [ .

The cond i t ion is c l e a r l y necessary. To show that i t is s u f f i c i e n t , w r i t e

f = t m g(1 - t ) n wi th g ( t ) ~ 0 fo r t ~ [ 0 , I ] . By (d), g > 0 and by (c) , f > O.

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NOTATION INDEX

(References are to the pages on which the symbols are def ined) .

group of complex numbers.

Q group of ra t iona l numbers

IR group of real numbers

T group of complex numbers of modulus 1

I group o f in tegers

GROUPOID THEORY

x, y, z . . . . elements o f the groupoid G

u, v, w . . . . un i t s o f the groupoid G

S,T . . . . or s , t . . . . G-sets or G-maps

d (x ) , r ( x ) 6

d (s ) , r ( s ) I0

xs, sx 10

u.s 10

AB, A - I 6

G O , G 2 6

G u, G v, G u G(u) 6 v ~ [u] [A] 35

G E or G I E 8

S 1 × G 2 122

G 1 m G 2 122 G(c) 8

Gx~A 8

G a 73

H\G 75 G n 11

Cn(G,A), Zn(G,A),Bn(G,A), Hn(G,A) 12

£G(A), £(a) 12 Ext(A,G) 13

Ko(G ) , D(G) 131

~b' gc 33

gn 14

cn(g ,4 ) , zn(g ,~) , Bn(g,~) , Hn(g,~) 15

EXAMPLES

CAR 129

GICAR 130

O n 140

O n 141

HAAR SYSTEMS AND MEASURES

{lm} 16 {(~2) x } 17

v, ~2, v-1 22

VO 24 D 23

[4 z4 ~(. ,s) z9 &(- ,s ) 29

COCYCLES

c one-cocycle

Min(c) 27

R(c), R~(C) 36

RU(c),R~(C) 36

T(c) 37

Rl(C ) 45

T-valued two-cocycle

15

3ROUPOID ALGEBRAS

Cc(X ) 16 B(G) 61

Cc(G,~ ) 48 B(G,~) 61

f , g func t ions on the groupoid G

h func t ion on the u n i t space G O

GROUPOID ALGEBRAS (con t i nued )

f . g 48

f * 48

s f , f s , s * f 62

hf 59

h s 64

rlflli,,~ , llelli,d ' I1flli 50 [Ifll 51 ]lfllred 82 C * ( G , d ) , C*(G) 58

C* (G,~) 82 red

~ ( B ) , g ( B ) , qj~(B) 104

s a 104

r (~ ) 112

156

f • ¢, ¢ • f , ,@ • f ,

< f ' g> B' < f ' g> E

Ind,~, IndM

f • ~ 77

78

82

SUBJECT INDEX

(The f i r s t reference is to the page of these notes where the

expression is defined ; the fol lowing references are to

a r t i c les where a simi lar notion appears ; they are intended

only to serve as a guide to the subject ; standard references

to C*-algebra theory are [1~ , [64 ] and [60]).

Almost invar iant set 24, ~1] 274

amenable groupoid 92, ~1] 354

amenable quasi- invar iant measure 86

ample semi-group 2O, [2 119 ample semi-group of an abelian sub C~-algebra

approximately elementary groupoid 123

approximately f i n i t e groupoid 123, [5~

asymptotic range 36, [31] 317, [49]

i04,[ 2]

Borel G-set 33

bounded representation 51

C-bundle 11, ~9110

C-sheaf 14 C*-algebra of a groupoid 58, ~4] 35

Cartan subalgebra 106, 135, ~ i ] 335

coboundary 12

cocycle 12

cohomology group 12, ~6] 467

continuous G-set 33, 38

convolution product 48, ~5] 624

Cuntz algebra 145, ~5] , ~0]

Cuntz groupoid 140

Cuntz inverse semi-group. 141

Dimension group of an AF groupoid 131,

dimension range of an AF groupoid 131

d is jo in t union of groupoids 122

domain 6,10

[52] r • L27] 25

Elementary groupoid 123

Elliott group 132, [27], [25]

158

energy cocycle 116 energy operator 115 equivalence re la t ion 7, 17, 22, 34, ~1] ergodic measure 24, ~ I ] 274 essent ia l l y pr incipal groupoid 100 extension 12, ~6] 128 extension groupoid 73, ~4] 105

F in i te idempotent element 131 f i n i t e type groupoid 131

G-bundle I I , [79110 G-map I0 G-module bundle 11, [79] 10 G-set i0 g-sheaf 14 gauge automorphism group 129, [8] 227

Glimm groupoid 128, [35] ground state 27, [65] 98 group bundle 7, [79] 8 groupoid 5, [44]3, [53], [611256, [79]

Haar system 16, [68] 27, [77]2 homomorphism 7, [44] 4 horizontal Radon-Nikodym der ivat ive 29

Induced measure 22, [31] 293 induced representation 81, [63]

induct ive l i m i t of groupoids 122 i n f i n i t e type groupoid 131 invar iant mean (of a measure groupoid) 91, [83] 30 ~nvariant measure 27, [31] 293 Inverse semi-group 20, [52] 2 invo lu t ion 48,[75] 625 i r reduc ib le groupoid 35 Is ing model 126, [33] isotropy group 6

KMS condit ion 27, [73] 63

159

Measurewise amenable groupoid 92, [81] 354 minimal groupoid 35, [24]7 modular function 24, [44114, [311293

Non-singular G-set 33, [51] normalizer of an abelian subalgebra 104, ~11332, [17]

Orbit 6

Partial isomorphism 14 physical ground state 27, [651100 principal groupoid 6 , ~ i ] product of groupoids 122 product type cocycle 128, [9] properly ergodi¢ measure 26, [61] 278

Quasi-invariant measure 23, [31] 291

quasi-orbit 26, [5~ 447

r-discrete groupoid 18,[31] Radon-Nikodym derivative 24, [31] 293

range 6, I0 range of a cocycle 36 reduced C*-algebra of a groupoid 82 reduction of a groupoid 8, [44] 3 regular abelian subalgebra 104, [31] 332, [17] regular representation 55, [45] 54 re la t ive ly free action 21 representation of an inverse semi~group 143, [1] 363

representation of Cc(G,~) 50, [75] 626 representation of G 52, [7~ 626, [45]47

o-representation 52 a-regular representation 55, [45] 54 saturation of a measure 25 saturation of a set 35 semi-direct product 8, 96 s imi la r i ty 7, [6~ 259 skew-product 8, 93, [31] 315, [53] suf f ic ient ly many non-singular Borel G-sets 33

160

T-set of a cocycle 37, [13] 152

topological groupoid 16, [26] 137

transformation group 6, 17, 22, 34, [24]

t rans i t i ve groupoid 6, [75]

t rans i t i ve measure 26, [61] 277

type I groupoid 27, [36]

type I , I I 1, I I , I l l quasi -orb i t 27, [55] 447

Unit 6

unit space 6

Vert ical Radon-Ni~odym der ivat ive 29