Closed Id& in C(X) and Related Algebraic Structures

by

Ross Stokke

A thesis

pte~enteù to the University of Manitoba

in partial futfilment of the

reQuitements far the d e p e of

m e r of Science

in

Mathematics and Astmmomy

Wipeg, Manitoba, Canada, 1W7

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Abstract

Giwn a topological space X, the ring C(X) of continuous rd-valueci fwictiona on X

is endowed with what is d e d the 'uniform mefric'- The c l a d ide& of C ( X ) in this

metnc are of much interest, and a new, p d y aigebraic characterization of these ideds

is provided. The resuit is applied to describe the real maximal ide& of C(X), and to

characterize s e d typea of topological spaœs. A Calgebra is an archimedian lattice-

ordered algebra d d y telateci to C(X). z-id& in 4balgebras are examineci, and as an

application to this studs several conditions equivaient to regularity in a Cdgebra are

obtained. A d o r m xnetric may also be plaeed upon a hlgebra, and in this metric

the clcsed ideais of a @-algebra have received a fair a m t of rePesreh attention. We

give necesmry and dc ien t conditions to ensure that an ideal of a @algebrs is c l d ,

and for two broad Çlasses of hdgebras show that tbese conditions are quivalent, thus

generaiizing our characferization h m the C(X) case.

1 would iike to thank my supervising ptofesso-l:, Marloo Raybinn, for ail of his heip and

encouragement during the preparatian of this thesis. Dr. Rayburn is an enthusiastic,

knowIed~abie, al- entertainhg man, and 1 wi l i miss our discussions immensely. 1

d d a h like to thank Rofe580r Grant Woods for his inspired teacbing, bis kiodness,

and the generous interest he continues to a h o n in my mathematicai dedopement. I

am indebted to Profe580r Melvin Henriksen for initially q g s t h g this project, and for

later taking the time to O@& me bis &vice and encouragement. F i y , 1 muid iike to

thank (the enchanting) Anna Fbbertson, my mother, fathet, and sister for th& love and

support on which 1 knuw I can depend.

Contents

L Introduction 1

1.1 Introduction and Org80i2ati011 of the Thesis . . . . . . . . . . . . . . . . . 1

2 An Introduction to Rings of Contirmous Wctions 3

2.1 TheRiiigC(X) ......................... . . . . - . - . 3

23 The s t a d e c h c~mpactïhtion . . . . . . . . . . . . . . . . . . . . . . . 5

3 C1oeed Id- in C(X) 8

3.1 C i d Id& end Strong Divisib'ity . . . . . . . . . . . . . . . . . . - . . 9

3 3 Spaces X For Which Every Countab1y Generated Ideal of C(X) is Principal 15

3.3 W&y Lindelaf Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4 Intduction to *figebras 23

4.1 Prelimiiiaries . . . . . . . * . . . . * . . . . . . C . , . . . . . . . . . . . . 23 4.2 The Heoriksen-Joh.nson Representation Threm . . . . . . . . . . . . . . 27

4.3 UnifimnlyCIased&Algebras . . . . . . . . . . . . . . . . . . . . . . . . . 29

5 z-Id& in A1,Algebras 31

5.1 z - I d d d z - F i l t m .............................. 32

...................... 5 2 Prime z-Ide& and Prime z-Ffltrs 35

........................... 5.3 N,andPrimez-ideab.. 38

.......................... 5.4 An Application: P-*bras 39

6 Cl& ideah of @-Algebras 45

. . . . . . . . . . . . . . . . . . . . . . . . 6.1 CiOgedIdealsandPrfmeId~ 46

. . . . . . . . . . . . . . . . . . 63 z-Id&, 2-ide&, and Strong Divisib'ity 47

. . . . . . . . . . . . . . . . . . . . . . . 6.3 Fen~rAlgebrasofUgebm. . 54

Chapter 1

Introduction

1.1 Introduction and Oiganization of the Thesis

This work divides ifseif naturally hito two parts. The Brst part of the thesis is compnsed

of chaptexs two anà t h , which tdœ place in a C(X) setting- The linal chapters make

up part tno of the thesis, in which the structure under considetatian is a type of lattice-

O f d d algebra, d e d a ??-aigebm

Begimiq with a topological space X, the ring C(X) of ali co~~tinuous real-valued

functons on X is e n d 4 with what is d e d the W o m metric'. In [Al] the aigebraic

notion of a st&y divisible ideai was examined by F. Azarpanah, and in the third

chapter of this thesis, we emphy this concept to charactetize the (UILifdy) dosed

id& in C(X) as preciseiy the stmngly divisible z-id& in C(X). This resuit is then

appiied to describe the d m&mal ideah in C(X), and to characterize dcompact and

peudocompact spaœs. Beyond thls ae show that e apace X for which every countably

generated i d d in C(X) is principai is neœssady finite, a d provide two new C(X)-type

characterizatims of weakiy Liideldf spaces. The th- of rings of continuous tunctions

which is requlred for chapter three, is briefiy outIined in the second chapter of this work.

CHAPTER 1. INTRODUCTION 2

A hslgebra is an archimediau ia t t ïdered aigebra with pmperties much Iike C(X) , where X is a topdogical spafe. A unifam metric may aLPo be defined on a Webra ,

and our priilcipai motivation in shtdying higebms is to g m d b e our characterization

of the closed id& in C(X) to a *algebra setting. In the final chapter of this eqxsition,

neees~ary and dcient amditim Tot an ideal to be elosed are obtained, and it is shocmn

that foc two relatively extensive dasees of h i g e b m , these eonditiom agree; thus in

these cases, our attempt to characterize the closeù ide& of a W b r a is successflil-

As in the C ( X ) case, the bigebra resuits described above emphy the notion of z-

ideal, and in chapter fi= we take the opportwiity to examine in sow depth the d e of

z-id& in *dgebras. Using t-ideab, we derive and extend some of the results found

in [HJI, and a b y show that z-ideals in *algebras are natural and useful. As an

application to our study of z-ideals, we consider what we c d a P-aigebra - a ip-algebra

in which evay prime I-ideal is rmxhai - a generalization of C(X) where X is a P-space-

Several quivalent conditions fm a @-algebra to be a P-algebra are given, for example if

@-algebra A is unifody clœed, then A is a P-algebra if and only if it is regular-

Chapter four provides a snrvey of the theory of O--bras to be used in the fUth and

sixth chapters of the thesis. The symbol Q wi l l be used to denote the end of a proof. W e

remark that attempts have been made to keep this work essentially self-contained.

Chapter 2

An Introduction to Rings of

Cont inuous hnct ions

The intent of this chapter is to pmvide a brief exposition of the terminology and notation

to be used in the ensuing portions of this work. Ali of the material in this chapter is

d-known and csa be f d without exception in the textbooks [GJI by L. Gillman

and M. Jerison, sed PWl by J. Porter and RG. Woods. A reader iinfiuniliar with the

study of rings of codinuous functions, wèo is perhaps looking for a sense of context, may

find that in the prefaces of the aforementioned books, and the survey article m] by M. Henriksen.

2.1 The Ring C(X)

It is shm in Chapter thrre of [GJj that if X is any topoiogical space, then there is

a T y c h d space, (that is a compktely regular EEausdorfE s-), X' such that C(X)

and ~ ( f ) are ring isomarphic. Unless it is expiicitly stated othernise, henceforth al1

tapological spaces wiii be assumed to be Tychomff.

For f E C o , the zero-set d f is Z( f ) = /'(O) = (z E X : f(z) = O), and

coz f = X\Z( j ) fs called the coze~o - set of f. The coilection of all zemsets of X,

Z(X) = { Z ( n : f E C(X)) is d d under finite union and onintable intersection;

co~l~equendy Z ( X ) is a lattice under the set contriinment relation C. A flter on the

Iattice of zemsets is d e d a z - fitter. Thus a z-filter is a coUection 3 of zereets of

X satisrying

(i) 0 e F9 (ii) 2-,Z2 E F, implies Zl n & E F, and

(ii) Zl E 7, Z E Z(X) and Zi c 2, implies 2 E 3-

A space X is cosnpletely regular (but not ngzssily Hausdorff) if and only if its zero-

sets comprise a base fot the cliosed sukts of X; equidently if and only if its cozefo~sets

comprise a baae fm the open subsets of X.

A hinction f E C ( X ) is a imit of the ring C(X) if and oniy if Z ( f ) = 0, and is a

divisor of zero if and only if Z( f) has mn-pty interior.

The f-g whkh is [GJ, ID] is used fiequently in chapter three.

2.1.1 Propoeitioa Lct j , g € C(X).

(1) If Z(f) a mghbdiood of Z@), tikn f ir o rnuuiple of g - that is f = hg for some

h E C(X).

(e) r f If 1 I 191' for some mi r > 1, then f Lc a mdtipie of g.

Note that the 8880Ci8tim f - Z(f) may be regardai as a qective mapping fimm

C(X) onto Z(X), and a?r such the f m notation is used. If A C C(X) and S c Z(X),

then Z[A] = { Z ( n : f E A}, and Z'[S] = { f E C(X) : Z ( n E S). An ided I of C(X)

is d e d a z - ideol if Z(f) E Z[l j ïmpiies f E I - that is 1 = Z'-[Z[Il]. It is easily

seen that maàmal id& in C(X) are z-fde81Se The folloaing theoran which Ir [GJ, 2.31

describes the relatimiship between id& of C(X) and z-fflters on X.

If M is a maxhnal ideal of C(X), and U is a z-ultrafilter on X, then Z[W is a

z-uitrafilter on X and Z'[U] is a maximal ideal in C ( X ) . Thus th= is a one-tcmne

correspondence between the maximal ide& in C(X) and the z-uitrafilters on X.

An ideal 1 in C(X) is ffied if n Z[q # 0, otberwise I is f ree. The hed maximal

id& of C(X) are precisely the sets

It is not difficutt to see that for each p E X, n Z[MJ = (pl, [GJ, 461.

AspsceXisC-emkdded(resp. C -embeddcd) inaspace Ycontaining Xiffor

every f E C(X) (resp. f E C ( X ) ) , there is a F E C(Y) (resp. F E C ( Y ) ) such that

FIX = f.

2.2 The stone-~ech Compactiflcation

In this section ne b r i e d d b e the ~ton&ech eornpectific8tion PX of a spece X in

the manner that d m a t benefit us in cbapter three. Details of the const~ction may

be found in chapter six of [GJI and chapter four of pw.

Adjoin to X one new point for esch hee maximal ided of C(X). Thus an index set

for the maxjmai ideais, denoteci PX, is formed; the maximi icleals of C(X) are precisely

{MP : p E BX). As noteù d e r the fixed madmal ide& of C(X) are {M'p : p E X),

hence X forms a ready-de index set for the fbteà maximal ideah of C(X), and for

p E X, lldP and Mp are uacd interchangeably.

PX is topologized in such a way that it is a compact Hausdd spaœ containuig (a

homeornorphic copy d) X densely, (that is PX is a compactïficatiou of X) , with the

additionai pmperty that

As a compactification of X , DX is unique in the saise that if 7X is any other com-

pactification of X in whieh X is Cembedded, then there is a homeomorphism h m

DX ont0 7X that &<es X pointwise- Eqivaienttly one might say that @X is the unique

mmpacfification of X such that C(X) and C m ) are ring isomorphic via the map

CCBX) -r C(X) : f r f lx. PX ia called the ~tone&ch corn-ficotion of X.

Evidently,

X is colllpcxd i f and only if X = PX.

Hence X ia eompact if and only if every maximal ideal of C(X) is M.

To condude this chapter we endeavot to define dcompactness and describe the

Hewitt realcompectification of X. For more on realcompact spaces, the reader is r e f d

to chapters five and eight of [GJ], a d chapter five of pw].

If I is an ideal of C(X), then for each f E C(X) , I( f ) will denote the member of

C(X)/I foi which f is a cosd representative. If M is a msdmal ideal of C(X) , then

C(X)/M is a tutelly ordered field containhg a copy of the d field R via the embedding

m p R - C(X)/M given by r - M(r). M is d e d r d or h y p e r - d accordhg as, with

respect to the above m8p7 C ( X ) / M I- R or C(X) /M contauis R properly. The f o h b g

theorem [GJ, 5.141 characterizes real ide& in temis of their correspunding z-uitdters.

Let vX = @ E PX : MP U rd), and give vX the subspaee tapoaogY induced by

PX. Every iùced maxhd ideal is reai, [GJ, 5-61, hence X c VX c PX, and X L dense

in v X . X ùi d e d dwrnpct if ewry fke maximai ideal of C(X) is hw-red- tbat is

It can be s h tbat (see for example chapter eight of [GJD UX is reaicompact and is

unique (up to a homeomorphism fixing X pointwise) amongst those dcompact spaces

containing X d d y wi th respect to the propery

Equivalently, VX is the unique rdcoaipact spce containhg X daisely such that the

map C(uX) -. C(X) : f o flX is a ring isomormphisrn. vX is called the Hewüt

~ m p u d i f i w t i o n of X.

A spirce X is paeudommp~ct if every continuous real-valueci continuous function on X

is bounded - that is C(X) = C ( X ) . In üght of pmperties (*) of flX and (**) of v X , it

is evident that

XiapsetLdOcOmpactif andonly i f u X = W

Altbugh this survey doea not exhaust the theary of rings d caitinuom funetions to

be used in this expaaition, we are now in a position to begin chapter tbree.

Chapter 3

Closed Ideals in C ( X )

BegInning with a (TychomfE) space X, a metric p on C(X) is defined by

It is clear that sequence convergence in (C(X), p) is d o r m convergence in the ordùisry

sense, and as sUEh p is d e d the u n i f m rnmieon C(X) and its induced topoiogy is the

uni- topology, or cn-toporOgar on C(X). For more intbrmation on the UILiform topology

the d e r is r e f d to Fm] in which p was inttoduced by E. Hewitt. In the sequel, d

topologicd promies of C(X) will be with respect to the uaifoam topology.

If K is a compact spaœ then the d o m topology on C(K) coincides with the

supremum n o m topology on C(K). It is weli-known that for compact K, C(K) with

the supremum nom is a Baiisch algebra. In conaagt, if X is an arbitrary space, then

C(X) is a manpiete topdogid vector spaœ in the uniform metrie, but is not in generd

a topologid algebm Indeed, in the absence of pseudocompactness, multiplication in

C(X) Ur not continuous, a fact essiiy derived from (3.1.6). ûur chi& con- will be with

the c l d id& of C(X).

I€ K is a compact space then the claaed id& of C(K) are precisely the intersections of

maximai ideab of C(K), [GJ, 401. In the case of an arbitrary space X, a characterization

of the cimeci ideals d C(X) wss achieved by Nanzetta anà PPlsnlt in w], howiever theu

th& description, the me qnaed above f a cl& ide& of C(K), where K is

compact), is highiy non-aigebraic in nature. In the next sectian a new, pmely aigebraic

characterization of the ebsed id& of C(X) ïs given, The d t ia then applied to

describe the red maximal ideais of C(X) and to cbm&erh pseudocompact speees.

Where there is overlep wïth pl, the prooh given are independent of, and perhaps

simpler than thme found in that gmper.

The principd t d to be used in this chapter is the concept of strong divisibiiïty which

was introduced by F. Azarpanah in [Ai] and used th- to chatacterize Lindel8 speces. In

section two, the results of 3.1 are used to show that a space X for which every countably

generated ideal of C(X) is priacipal is necessarily fhiite. The tlnal section of the chapter

contains a chatacterization of weakiy Loiderof speces in which a d c t e d *.pe of strongly

divisible ideal is emplqed-

3.1 Closed Ideals and Strong Divisibility

3.1.1 DeMion: An ideBl1 of a mmut4tsPe ring R is d t e d stnmgb divisible, ( id ) ,

iffwmery wuntableSZLb8et{a,, : n e N ) o f l h i s a n a E I and ambset{b, : n € N )

ofRsvchthatj"ea&n~N,ab, =a,, .

Thus I is strongly divisible if for every amtable subset C of 1, the elements of C possess

a cornmon divisor in 1. One imrnediateiy obames that any principal ideai is strongly

divisible, and any coufltably generated strongly divisible ideal is principal. The Brst part

of the fdowing thgRm is by F. Azaqmuù and is found in [Al]. Its proof is not long

and so for the d e af umpleteness we ch- to ioclude it.

3.1.2 Theorem: Let X be a Qchonof ~pace.

1. [Atqnaruh] r f 1 is a z-ideBI of C(X) mch Ulat Z[l j i s closcd under mntable inter-

P m f i 1. Let (f,) c 1. By Weiezstmss,

beiongs to C(X) and dearly Z@) = ('&Z(f,). Z[l j is CM under c-bk ùitersec-

tion, so Z(g) E Z[I] . I is LI z-ideai so g E 1. But for each n, g 2 2-l.,+, and thdm l+ n

1 fnl I IZn(l + f''3)913/2- By (2.1.1) , d f, is a multipIe of T(1+ ftPlg, hence each

f, is a multiple of g. Thetefore I is strongly divisible.

2. Let ( fn) C 1, so that {Z( f,) : n E N) is an arbitrary subset of Z[q. I is strongly

divisible, so t h g E I such that for each n, g divides fn. It is clear then that Z@) c El Z(fn). But Z[q is a z-filter containhg Z@), so E-l Z( fn) E Z[I ] . 9

PmoE 1. A maximal ideal M of C(X) is real if and only if Z[M] is closed under

count able intersection (2.2.1) if and only if M is strongly divisible (by the above t heorem) . 2. X is peudocompact if and only if v X = PX if and only if evay meXimal ideal is real.

O

If A ia a subaet d C(X) then A ssin denote its ( d o m ) dosure in C(X).

Pmoll- + 2. Let 1 be a closed ideal of C(X) . Suppose f E C(X) d g E I are such

that Z( f ) = Z@, . Far ewry positive integer n, let h. = [( f - L/n} v O] + [(f + l/n) A O)] - Then for eaeh n, Z ( k ) = f ' [- l /n, l /n], so Z@) = Z( f ) c intZ(h,,). By (1.1.1) &

h,, is a multiple of g, henee egch h,, E I . But 1 f - h,J 2/n -. O as n -. oo, so h,

co- to f t m i f d y (i.e. hi converges to f in the Morm t e ) . S i 1 is

c l a d it foUm that f E I , praruig that I is a z-ideal. It remains to prove that I is

stroxgly divisible.

Let ( fn) be a countable subat of I . By (3-1.2) , to pmve strong divisibiliw it suffices

to show that El Z ( f n ) E Z[I]. Let g = xf 1 f n j A 2 7 which by Weierstrass bebngs

to C(X). If for each n E N, gn = 1 fkl A 2-', then ît is clear that the sequence (g,)

converges to g uniformly. But for each n, Z@,) = flLl Z( fk) E Z[I], and I is a z-ideal,

w each g, E 1- I is c i d , so g E I and therefore c-, Z( fn) = Z(g) E Z[I] .

Earlier it wa9 clru'med that out characterization of cl& ideais in C(X) aas of a

pureiy algebrsic nature. Althou@ the definition of z-ideal given in bpter 2 may not

have seemed especially algebrak, [G J, 4 4 asserts that the foilowing algebraic condition

is necessary and diCient for an ided I in C(X) to be a z-ideal:

Given f E C(X), if th- exists g E I such that f belongs to every maiomal ideal

containing g, then f E I.

The f0Uowin.g extends Codary 4.3 of pl. We note that by a d-lai- theorem

of algebra, if 1 is an ided in C(X) such that C ( X ) / I is isomorpic to the real field R then 1 is necessarily maximal.

3.1.5 CoroLlary nie folloving un cpuivaknt for an ideai 1 of C(X) .

PmoE The quivalence of l., 2., and 5. is dear fkom (3.1.3) and (3.1.4), that 2. implies

3. is obvious.

3. =+ 4. If I C J , where J is strongiy divisible, then Zb[ZIJ1] is a strongly divisible

z-ideal (by 3-12), henoe dosed by (3.l.4), with I c J c Z+[Z[A] . By 3. 1 = Z-[Z[J] ,

henœ 1 = J . Thdore I fs a maximal strongly divisible ideal of C ( X ) .

4. =+ 3. Were I C J , wïth J cl& then J is stmngly divisib1e by (3.1.4), hence I = J by

4.

3. =+ 5. Suppahg I is not maximal, take M a maximsl ideal of C ( X ) with I properly

coatainedinM. If f ~1M\I,thenwedrUmthat

is a base for a z-filter A on X that is cl& mder countable intersection. To see this,

obderve that A C Z[W, Z[Mj a z-füter so 0 $ A. 1 is cl&, therefore strongly divisible,

hence by (3.1.2) Z[I] Is CM under countable intersection; it foUows that B is ci&

under countable intersection, w h c e A is closed under cwntable intersection-

Rom (3.1.2) Z6[d] AIS a stmnpiy divisible sideal, hence a c l d Meel of C ( X ) containhg

1. But f E Zc[A], f $1, so this containment is proper, a contradiction to 3. O

That an idd is mereiy strongly divisible does not alaie p t e e that it is dosed,

that is, strongly divisible ideah need not be z-id&. For example, if i denotes the identity

fimtion on R, then the principal, (hence stmngly divisible) ideal (i) is not a z-ideal in

C(R); se [GJ, 2-41.

The follOWiDg is a sllght exteasion of Theorem 2.1 of [NP]. A shorter though les

interesting proof d theh 5d implication is givea,

Proof 1. + 2. If X 16 pseudocompact, then the map C o - C(X) : f -.. flX is

an isometrïc isamorphism. Since PX is compact, by [GJ,SM] dosures of ide& of C m )

are ideais. 2. follows.

3. =+ 4. Let I be a stn,ngly divisible ideai d C ( X ) . Then Z[I] is CM mder

=table mersection by (3.1.2). But J = Z'[Z[Zj] is a z-ideal, with Z[J = Z[q c i a d

under coimtable intersection. Hem J is a strongiy divisible z-ideal, therefore c l d , and

Ic J . Thus4. f o h ~ 3 .

4. =+ 1. By 4., ewry maximal Meal is c l d , therefore reai by (3.13). Haice

VX = PX and X is pseudocompact. 0

We remark that it is clear from the above corollary and proof of 3. * 4. that if X is

pseudocompact and I is a strongly divisible ideal of C(X), then f = Z'[Z[I]].

W e now attempt to determine which prime ide& of C ( X ) are c i a d in the unifonn

topology. Eleesll that for each point p E PX, OP denotes the i d d consMing of ail

fiinctions f in C(X) for which daxZ(f) is a ndghborhood d p. That is, for each p E OX,

if p E X, OP is aiso denoteù Op, and may be describeci more simply as

For each p E PX, OP c W, (an obvious co~l~equence of the G&811d-KoIniog0~0ff Theo-

rem, [GJ, 7.31). Detaüs of these assertions may be found in chapter 7 d [GJ].

PmoE It is enou& to show that MP C v, so let f E MP and let E > O. Let

9 = [(f -E) vO] + [(f +E) ho!. Then Z(g) = T[-E,E]. Now by [GJ,7D] r(p) = O, where

/. : BX - R* is the (unique) continuous firnction b r n PX into R*, (the aie point

compactification of R), such that f lX = f. By [GJ, 7.121, Z(g) E Z[W] and therefore,

since P is a z-ideal, g E P. ut [/ og[ < E, hence f EV- O

[G J, 7-15] m s that for every prime ideal P of C(X), there is a unique p E PX such

that OP C P C W. It follows h m the lemma that ao non--al prime ideal of C(X)

is d d . However, as noted =lier, the ci& rwchal ideais of C(X) are pmhely the

r d ideais of C(X), i.e. MP b d d if and only if p E uX. Hence we have

A spaœ X ùi pseudocompact if and only if f l = v X . Hence ne get the following

extension of (3.1.6).

Note tbt for p E v X , the z-ideal Or ïs not dosed under coumable inteRection

and theteforet by (3.1.2), is not stmngly divisibIe, yet dP = &P is an i d d Thus, if I is

a z-ideai of C(X) such that is an ideal d C(X), it does mt fdow that I is stmIigly

divisible-

3.2 Spaces X For Which Every Countably Generated Ideal

of C(X) is Principal

As an application to the above, in this section it is shown that thoae spaces with the

pro- of the preceding title are precïdy the finite (Tychonoa) apaces. The result

is perhaps 8inpdsmg given that F-spoces - spaoes X for which every ânitely genemted

ideal in C ( X ) is principai- may be non-discrete and of ~nc0mt8bly mfinite cardinality,

for example BN\N, [GJT 14-27. Our theorem hinges upon the f o m obsenratiou.

3.2.1 Proposition: nie foIIoving are quivcrlent for a oommtâatine ring R:

1. Epery idd in R is stmngly dittisible.

2- Eu- c o t ~ n t a b l ~ genaokd idcd of R is pincipal.

PmoE 1. + 2. That strongly divisible, eountably generated id& an ptinciH is obvi-

OUS*

2. + 1. Let I be an i d d of R and let {h) be any countable subset of 1. Then

J = (a, : n = 1,2, ...) the ideal generated by {a,,) is principd, say J = (a), where

a E R. But a divides each h, and since J c Il a E I . Hence I is strongly divisible- O

In [Al], Azaqmah calls an ideal I of a c01ILrnutative ring R divisible if for any hite

subset {al, ..., h) of 1, there is an a E I , aud a subset {bit ..., Ri) of R such that for

1 < i n, obi = e. In other words, 1 is divisible if every b i te subset of I has a divisor

in 1. Just as a b e t the folImhg is true-

3.2.3 CotoLlary: X k an F-spaee if and ody if e v q fdeol ofC(X) is divistvistble.

The foRowing is a special case of a more gieneral resuIt of De Mana, see pl. Our own

ptoof is provided.

3.2.4 Lemma If X is cumpuct, then üie ideal I = kSO,, , w h S P o zero-set of X,

is muntiablg gmmted.

P-fi Let S = Z@), where g E C(X), and consider the functions gn = [(g - l/n) v O] + [(g + l/n) h O], n = 1,2, ... . Then for each n, Z(g,) = gl-[-lin, l/nj, and hence,

S C int(Z@n)). It f0110~s that each g . E 1, d & Z(gn) = Z@). Non t h f E 1, so

that S C intZ( f). The inciie9sing -ce of open sets {X\Z(gm) : n = 1,2, ...) aivers

the compact set X\intZ(f). Thdore a poeitive integer k can be chmen such that

X\ntZ(n c X\Z@,), hence Z ( l ) ïs a neîghborhood of Z(g& fmn which it foliows

that gk divide /, (2.1.1). Thus I is genemted by bn : n = l ,2, ...). O

A topologicd space is a P J ~ ~ Q if each of its -sets is open. (For more on P-spaces,

sec [GJ, -1). W e will use a result fmm [GJ, 4Kj in the next proof.

PrcmE If every countably generated i d d of C(X) is principal, then by (3.2.1) , every ideal

of C ( X ) is strongly divisible, and consequent1y, X is pseudocompact (3.1.6). Suppose for

nowthat Xismmpactmdlet SbeazercmtofX. Bythelemma,theideaiI=nes4

is countably generated and therefom principai by hypothesis, so tsk f E 1 such that

(1) = 1. Clesrly then, Z( f) = S, but since f E I, it is dso tnie that S c intZ( 1). Haice

S = Z( f ) is open end therefare X is a P-spaœ- But compact P-spaces are h i t e [GJ, 4, so X is finite. If X is BgSUrned oaly to be pseudocompact, then C(X) is ring isomorpbic

to C m , (/3X is the Stondmû compsdlficatiim d X), so every coutably generated

ideal of C@X) is principal, and by the above argument flX is hite, whence X is finite.

Conversely suppae X is ûnite, and let I = (fn : n = 1,2, ...) be the m t a b l y

generated, ( pmper, non-trivial) ideal of C(X) generated by { f, : n = 1.2, ...). By the

blanket 8ssumption that ail spaces are Tychcwoff, X is discrete (and compact). Thetefore

the -set S = nZ[q = &Z(tn) is a non-empty pmper su- of X, and so the

characteristic fwiction, cail it f, on X\S is a non-unit in C(X). Noa Z(f) = S, so for

each n, Z(f) C Z( fa), and dmly f fn = fn, hem! ( f ) > K. On the d e r hsad, using

a compactness argument (simjiar to the one used in the proof of the previous lemma), it

is easy to see that for some n, Z(fk) = S. Define g to be ff + ... + fn; then g E 1,

and Z@) = Z( f ) . Mine

X is disate, SQ h E C ( X ) , and f = hg E 1. It foîlows that (f) = 1. O

3.2.6 ComIlary: E v m ided of C(X) is &ed if and only if X is finite.

PmoE Cbeed id& are strorgly divisible, which together with (39.1) and (32.5).

provides neœssity. If X X iste, then by the abave every countably generated ideal of

C(X) is principal, hence every ideai of C ( X ) is stnwgly divisible. As a disrete spaee,

X is also a P-space, therefore evety ideai of C(X) is also a z-ideal, [GJ, Alj. Thus every

ideal of C(X) is cl&, by (3-1.4). O

3.3 W d y Lindel% Spaces

In F. Azsrpanah>s paper [Al], the notion d strongiy divisible id& ïs used to cbmctmbe

(Tychonon) LinàelX speces as téose spaces X such that ewry strongiy divisible i d d of

C(X) is M. Hkre we give a similar -ion for weakly Linde18 spar#s-

RecaU that a space X is mcaLty LindeIof Kevexy open cover of X contains a countable

sübfamiy whme union is dense in X, pq. A coktionC of subsets of a space X wiU be

said to have the sbong oountable intersedion p p r @ (SCIP) if the intersection of =y

couutable subfamly of C ha8 nmempq interior. It is dear that a space X is Lindelof if

and only if any dectim of closed subsets of X with the coutable intersection property

has nonenpty intersection. The foUowing is the d o g u e fot weakly Lindel8 spaces. A

subset of a topobgical space is rogulm dased if it is the cbsure of an open set.

Pmd: 1. * 2. Suppoeing 2. is fdse, b C be a family of clos& subsets of X

with SCIP and n C = 0. Then {X\C : C E C) is an open cover of X. If X is weakly

Lindelof then there is a countable subfamily (Cm) of C such that d(U X\C,J = X. But

tben X = d(UX\C,) = d(X\nCn) = X\int(nC'), end thdore int(nCn) = 0, a

contradiction. M o r e X is not weakiy Lindelof.

3. 4. If U is a family of open subsets of X with SCIP then {d(U) : +Y é U) is a

f d y of ~egular c l d subets of X with SCIP, hence by 3., n{d(U) : U E U) # 8.

4. 1 Suppoeuig 1. is faIse let U = {U' : a E A) be an open cover of X such

that t h e ~ o n d w ~ k a i b f s m i y d U i a d e n i l e h X . Let D = { D c A : D is

count8b1e). For esdi D E D kt VD = X\dbaa Va]. Then each Vj is open and by the

hypothesis on 24, each VD # 0. Let V = {VD : D E D).

For let (Dn) be a countable s u b f ' y of D. Then D = U Dm E 2, and

But VD is open, nonempty aud thdore M(& V&) # 8, proving the claim. But

That 2. is quivalent to 5. ia immediate. 0

3-3.2 Defùûtion A st7vnglp divisible idad I o/a comtnutatt*ve ring R annprised entid;y

of ditdom of z m miU k d l e d neighborIiood brongly divisible, or dtnplg nsd.

Recdl fmm chapter 2 that a member / of C(X) is a divisor of zero if and cmiy if Z(f )

has non-empty interior. Thus an i d d 1 d C(X) ie nsà if given any countsble subfamiy

(A) of I there is a g E I and (k) c C(X) such thst for each II, fn = gh, and Z(g) has

non-empty interior,

3-33 Lemma Let X be QcirOnofi 1 a z - idd of C(X). Then the follolping atz

equivalent.

1. r ia nsd.

2. 2[4 has S C ' and is closed under mntcrble intersection.

PmoE The proof is similnr to the p d of (3.1.2). 0

3.3.4 Theorem The following are equiuutent for a @chonoff space X:

1- X 69 ~t?akl( Lindekif-

2- E o ~ nsd ided of C(X) is &ed.

PmoE 1. * 2- Suppose X is tKeakly Linda, 1 a nsd ideal of C(X)- Then, given any

countable subset (f,) of I there is a g E I and chn) c C(X) such that for each n, fn = ghn,

and intZ(g) # 0- Thdore, intZ(g) C Z@) C & Z( fa), h e m intn Z(fn)j # 0.

Thdore Zm bas 80 by (3.3.1), n Z[q # 0, that is I is hed.

2. =, 1. Let A be a collection of -sets (basic ci& in Tychonoff X), of X with

SCIP. Let 8 be the coilection of ail corntable ideisections of members of d. Then B is

a base for a z-6iter 3 M X which has SCIP and is c l d mder coutable intersection.

Thedore Z*[q is a z-idd and Z[Z- [a = F has SCIP, and is cl& mder anintable

intersection. By the lanma, Z-[A is nad. Hence, by 2., Z1-[fl is Bxed, so n f # 0.

But A C F so n A # 0, and X is weakly L'mdelOf. O

As obserwd by Professor H e d w e q a comllary to the above is the following theorem

which is part of PW, 5-11].

3.3.5 Corollrvy: W&g Lindcbf dm& P-spruzs ore Lindeboj?

PmoE Zemsets of almoet P-spoices have mnempty interior by definition. The result fol-

lows immediately korn the above and the Azaqamh eharanerizatian of Lindelof spsces,

which was quoted at the beginning of this section. 0

As a Bnel result we give mdher C(X)-type diaracterization of neekly Lindelof spaces

using id&, a sort d ideal whoae C(X) properties have also ôeen studied ex-

tensively by F. Auupariah.

3-38 D e h i t h An idcol 1 of a ring R t said to b essentid in R if ü m&b euery

nol~tn~uàal ideal of R non-&Wall&

Azarpanah sbanied that an i d d 1 of C(X) is essential in C(X) if and only if n Z[q is nowhere dense in X, i.e. if and only if ad(n Z[q) = 0, [ml. We use this theorem to

prove the following:

Pmofi 2. =+ 1. If C(X) contains no fme id& then X is compact, hence d y

Lindelgl, so we n e e that C(X) conteins free ide&. La F c C(X) such that U =

{X\z(f) : f E F) ls a @sic) opencoverofx. Let I = (f : f E F) be the ideal in

C(X) generated by F. If I = C(X) then, in the foilowing argument repiace I by aqy fh

(proper) ideal in C(X). U amers X so 1 is is and thezefore by 2. tbere is a m t a b l e

subset N d 1 such that E =< g : g E N > is essential. 1 is generated by F so for each

9 E N t h e aist fi ,..., fnb) E F BILd hl, ..., h,,@) E C(X) such that g = ~ ; ~ ) h ~ f ~ .

Hence for eachg E N, Z(g) 3 n;(o1Z(fL)- N m kt M = {fr : 1 5 k 5 n(g) ,g E N).

Then M is a coimtable su& of F. But E is essential, and is generated by N, so

1. * 2. Suppose X is weakly Lindelof, and I is a fkee ideai of C(X). Then n Z[I] = 0,

so u{X\Z(f) : f E I ) = X and therefore there is a countable sukt N of I such that

Chapter 4

Introduction to a-Algebras

In this chapter ne provide the definiticm of a eaigebra, and describe the Henrüuen - J o h n Representation Thw,rem by mesiur d which every *aigebra may be regded

es an aigebra of extended a - d u e d fimctiaas on a compact space. W e Ke give several

examples of -bras and indude mme of the theory that will be used in the nninininp

chaptm of this aork. Although atternpts are made to keep the W b r a portion of this

thesis d d y self-~~btfained, for furcher Tnfbmation the intaested d e r is teferred to

the seminal n a k on Wdgebras by M. Hemïksen and D.G. Johnsm [&Il and the siirvey

articles [H2], [H3] by M. Henrilisen fot an essentially eomplete history of the subject-

An 1 - dg& is an algebra A over an ordered fi& K which, under a partial ordering

2, is a lattice which satistles

1. a ~ b i m p l i e s a + c ~ b + c ,

2. a zOandb10, impll iesab~0, and

3. a 2 0 and a 2 0 implies au 2 O.

f o r a , b , c ~ A a n d a ~ K .

An I-aigebra A is saïd to be archimedian if, for a, b E A, a = O whenever nu 5 b for

aü integers n. A reai archimedean f-algebra with an identity is d e d a Wg&a; a

ih-algebra is necessarily commutativel [BP].

Simple exampIes of 4kalgebras are the trivial @-aigebra (O) and the field of real

numbers R C(X), the ring of di eontinuous mai-valued bctiom on a topologid space

X, is redy the prototype @-algebra. hdeed much of the acisting researeh on @-aigebras is

devoted to finding char~cterizations of C(X) for various kùYis of spaces X, an endeavour

which, in its most g e n d seme has yet to be completed, [a], [81. Interesthg examples

of O-aigebras whîch cennot be represented as C(X) for any space X, are the respective

algebras of Lebesgue measurablel and nd Baiieons on the reai line 8, [HJ, 5.11. W e

note that these aigebras taken modulo theh respective ideah of hnctions th& vanish

almost eve~ywhere are also ibaigebm, (definitions of Lebesgue meamrable, and Baire

functioru m y be found in mcmt books on meciarue and integration, aee for example wu]).

A resder interested in seeing more exotic examples of a-algebras can tind severai in PJ. If A is an t-algebra, then A+ = {a E A : a 2 O). For a E A, let a+ = a v O,

a- = (-a) v 0, and [al = a v (-a). Then a+ a- = 0, and

(i) a = af - a-, and

(i) la1 = a+ +a-.

If A is in fact an f-ring, then

(i) a2 2 O for esdi o E A, and

(iv) /dl = laIl bl for al1 a, b t A.

Roofs of these assertions may be fomd in BPI, though the reader is waswd that there

the authors define a- to be -(-a) VO.

If A and B are reai 1-algebras then a mapping 4 : A -, B which is both a lattice

If A is a nontrivial red f-aIgebra with identity 1, then the mapping t - r 1 is an

l-mammorphism from R into A; thus R is considemi a su-algebra of A by i d e n t m

r w i t h r - 1 .

is a subalgebra of A, anci is d e d the splwebm of bounded elements of A.

By an ùkd in an l-eigebra we SM mean a propet aigebra ideal. If I is an ideal of

1-algebra A, then it is d e d umvezr if whenever O 5 o b, and b E 1, then a E 1. An

ideal I in an 1-aigebra A is d e d an 1 - ided (or an ahlutely con= ideal) if a E I

whenever a E A, b E I anci [a[ Ibl.

If 1 19 an 1-ideal of the 1-algebra A, and a E A, then I(a) wiU denote the image of a

under the airionical hornomobphim b m A onto A/& that 19 I(a) denotes the residue

c b of a mddo 1. Thus I denotes both the 1-ideai and the candcd map A -, A / I .

Given a convex ideal 1 of an 1-aigebra A, define the relation 2 on A / I by sayhg

I(a) 2 O if there exïsts z E A such that z 2 0 and I(a) = I(z).

Under this relation A/I becanes a partïaüy ordered algebra, (thoee interesteci are refend

to [G J, 0.191 for the d a t i o n d a partiaily ordemi ring, and [GJ. 5.21 for details of the

assertion jnst ma&). Hendorth any reference to the order of a fsetor algebra will be to

the order as debed above. The importance of Lideais in l-aigebras ia apparent h m the

following themm which is [GJ, 5-31

4.1.1 Theoreun The follotlning conditions on a a m v ~ i&d I in an 1-*eh A are

equiucJent.

(1) 1 iP un l-i&!d.

(2) z E 1 i m p k 121 E 1.

(3)2,y E I impliesz~y EL

(1) I(a v b) = I(a) v I(b) . (5) I(a) 2 O if and onIy if I(u) = I(la1).

We note that in a part iaI iydd aigebra, if a v b exlsts fm all a, 6, then a ~b exists,

and a ~b = -(-a v -b), [GJ, 0.191. Thua K I ïs an Lideai of the 1-dgebra A, then A / I

is aiso an t-aigebra Note that by condition (3) 1 is a sublattice of A, and by condition

(4) the canonid mapping fkom A ont0 A / I Is a lanice homomorphism. Thus,

Let A be a -bra The cdection d aü m&md Eideals in A is denoted M(A).

For a E A, let

M(a) = {M E M(A) : a E M).

It can be d e d that the cuüeioa (M(a) : a E A) Is a base for the c i d sets of

a compact Hausdd topoiagy on M(A), caUed the Stone or hdI-kenrel topoiogy on

M(A); M(A) Is aleo r e f d to se the mozimoll-idaal spaee of A. Fa ddaiis the d e r

is referred to [HJ].

With regards to rn- 1-idcals for reférence w e quote the fo11owing fects whîch are

1.6 and 1.7 d M. (i) If M is a maximal I-ideal of O-elgebra A then M is a prime ideal, and A/M is a totdy

ordered f-Wb= nithout mn-zeto dhriso~ of O.

(ii) A maximal 1-ideal of a @-algebra A need mt be maximal a3 a ring ideal of A.

If M is a rmchai 1-ideal of A, then the tdally ordered algebra A/M contains the

real6eld R as a subfield via the embedding map R -. A/M : r u M(r 1). M is d e d

reai if A/M = R, otherwiae M is CM hyper-ml.

W e now tuni ota attention to describing the representation theorem.

4.2 The Henriksen-John Representation Theorem

Let = RU {Ioo) denote the tftn,poiat compactification d the real field R For a

compact space X, let D(X) demte the set of al l cmtinuous functions / : X + y R for

which

R(n ={zEX: f(z) ER}

is a dense (necessariy open) subset of X. The elexnents of D(X) are d e d e=dended

(d-vducd) finctions. Beginning with iùnctions f, g E D(X), and X E R, the functiom

Af, f h g , and f vgdeânedpoinfftrisearecEear1y also in D(X). Ifthereare functiom

h, k E D(X) satisfying

for d t E R(R(n n'Rb), then h and k are called the sum and product of f and g, and we

write h = f +g, k = fg. Note that since R ( f ) n 7Z(g) ùr deme in X these operations are

uniquely defined. Whüe A f , f ~ g , and f vg always exist in D(X), [HJ, 2-11 provides an

example where D(X) is cloaed d e r neither addition nor multiplication, Indeed [&Tg 2-21

easerts that D(X) is an aigebra if and only if dense a m m e t s of X are C-bedded in

XI

W e can mia state the representation tbeorem which is [HJ, 2-31.

Although it ui not our intention ta prove (4.2.1), (the interested d e r is referred

to [HJl for the proa), it wi l i be to our advantage in Chapte. six to han the precise

represeatation of an dement a E A, as an extendeci function on M(A).

To each a E A, a functim B : M(A) -. IR is associated as foliows.

if a E A+, and M E M(A), take

qM) = in f {A E R : M(a) A),

(where X is identifid with M(A - l), and inf 0 hr understd to be +cm).

If a E A is mb'rtrary, let

Z(M) = F(M) - F(M).

Note that since a+ A a- = û, M(a+) A M(a0) = O. But, as observed earlier, A/M is

totdy orderd end hence M(a+) = O, or M(ao) = O. Thus F(M) = O or F(M) = 0,

and Z is welldehed. Let denote the set of extendeci functions {a : a E A)

W e remark that the theorexn is proved by showing that as defined above bdongs to

D(M(A)) aiad a - Z is an l-embedding of A into D(M(A)).

Shoald expllcit consideration of the COIlStNction of O h m a be neœssary ne shaIl

refès to O as the ' H e e n - J o h n s o n ttpms-n of a ', and the d e r may wish to

review the above at that time. This hainrever shouid not be necesary until chapter six.

We presently outline some of the th- derïved in [HJ] nom the reptesentation the-

orem. The Brst d t is a Geifand-Kdmogomff type of characterization of the maximal

1-ide& in a Çalgebra A, [cf. GJ, 7-31, and wili be of fundamental importance to this

study. It t Theorem 2.5 of m.

4.2.2 Theotem A &et M of a O-olgeh A is n maPmd 1-ideol of A if and only if

thereisauniquez~M(A) suchthut

In lfght of the above, an elexnent d the msamal 1-ideai space M(A) will be written

as 'W, ot 'M,' if w e wish to view it as a maximal l-ideal, and simply '2' if we wish to

view it as a point of the tqobgical spaœ M(A).

Let A be a nontrivial real j-elgebra with 1. Given a, b E A, define

where, as usual, r - 1 and r are identifieci. The tileriftcation that p is a pseudometric on

A is mutine- p is defined on the Mvîai -bra in the obvious way. p iis called the

u m f m pseudometric on A and the topoiogy induced by p is d e d the unif' topologyogy

Henceforth aU refisences to tapological propertiea of an f -aigebm wi l l be with respect to

its unifixm topo-

If A is complete with respect to p, then A is said to be uni#mdly dosed Wd-known

examples of uniformly c l d @-algebras are R and C(X) , where X is any topologicd

space- Indeed many @-aigebms of interest, such s the respective algebras of Bsire and

Lebesgue functiom on the r d h e are uniformly cioseci, and oonsequently the properties

of ~Uormly chsed *aigebra are of especial intaest. W e outhe some of these properties

m; the fiRt is m, 3.21 and [BJ, 3.1.

43-1 Tharram A O-dgeh A is unif&. &ed if and d g VA* and C(M(A)) are

isomorphic

A Calgebra A is c l o d under h n d e à inversimifa E A, o 2 1, implies l / a E A. The

principal ideal of a member a of 4algebra A will be denoted (a), thus (a) = {ab : b E A).

The smallest 1-klea of A conteiniiig a niIl be denoteci (a)i and d be caïîed the I-pri&pd

W o f A g e n e m t d b p a , (esin[Bj).It iseasy toaeethat foranya~A,

(a)( = {c E A : [cl I Id[ for m e b E A).

W e condude the present chapter by stating the foliowing cbaracterïzation theorem

which is IHJ, 3.91. Note that an dement a of a Calgebra A of extendecl hmctiona is a

divisor of zero if and ody if a'(0) hm nuin-empty interior, (see page 86, paragraph 2 of

[aiI for de ta ) .

4-33 Theorem A hdgebru A à& isorntnphic to D(X) f' soine compod space X if

and only $

(1) A k un~ormly closed, and

(2) I ~ u E A , h e f t h e r a às adi tJ igorofze~~, or(a)* = A .

Chapter 5

The main objective in out study of @-aigebras is to generalize the characterization of

c l d ideab of C(X) es found in ehspter thmet to a +aigebta settïng. The notion of

a z-ideal in C(X) was centrai to au e a r k th- as it will be when ne attempt to

describe the c h a i C-ideais of +aigebras. Because z-id& are cnrcial ta the study of

rings of conümms functions, rather tben simply using them fa the apecific pirpase

of describing dosed bideais of 0-*bras, we taie this opportunity to more b d y

Bcamine th& d e in *algebras. Usiog tideaia ne -and upon m e of the reaults

found in [Hq, thus Ui~sftating the use of sideais in this more general context. Beyond

this, as sa application, ne conclude the chapter with a study of what we call P-aigebras

- a-algebras in which prime Gideals are maximai - a generalization of C(X), where X is

a P-space- We note that that 2-Meals in the erra more g e n d settings of commutative

rings and paalallyadered rings were studied by G. Mason in M. There, some of the

resuits obtaineù in thie chapter may aleo k f d , (unbeknownst to the author at the

the when this work was done!), though we mwrk that out notation is rather difkent

h m Mason's. Due to the fsct that ne sh4R remain oniy in the world of C algebras and

therefore have acasa to the H-Johnson representation theore., in those places

where there is overlap wïth [Ml, the resuits that f o h are sometimes slightly stronger

than téose found th--

5.1 z-Id& and r-Filters

Note that since M( f ) > M(g) if and d y if M ( f ) = M(/g), '3' may without 1m of

generality be replaced by '=' in the above definition. As noted in chapter 3, if X is a

topologid space, then the abme definition of z-ideal agrees with the usual notion of a

z-ideal in C(X), [GJ, 4A].

In it is pmved that the maximal Gide& of A are preasely the sets

It foliows immediately that fm each f E A

These subsets of M(A) wiü be shown to play a nimiinr d e to that of zeresets of a

topologid space X. Indeed, if X caae a compact topoIogicai space, and M(C(X)) the

rmdmal ideal space of C(X), then fot f E C(X) , the sets M( f ) and Z(J) are equal, (up

to the equivalence of X a d M(C(X))).

We note that M arn be regardeci as a SUtjective map M : A -r M[A] : f - M ( f ) ,

and as such the foJlowfng notation is employed. If S is a subset of A, then the coilcction

of subsetsofM(A), { M ( n : f ES}, wi i l bedenotedbyM[q. Hence M[A] = { M ( j ) :

j E A) and M(A) = {M : M is a mazimd 1 - ideal of A}; square and round brackets

distingurshthedineratce. IfSisamMdM[A1,t6enMt[S]={f e A : M ( f ) ES).

Using this mtation we see that

ris a 2-idcalin Aif andonlyif I=M-[M[q] .

As a Brst eoiincction betneen the zero-sets of a (ccmpIetely ngular) topologid space

and the ~01Eection M[A], observe that by the very defintion of the Stone topdogy on

M(A), M[A] compdses a base for the c i d subsets of M(A). A second connection is

the foiiowhg

5.1.2 Pmpœition: M[A] fimns a ùattice under set wnhinment.

Proofi Let f, g f A. Then

= (Z E M (A) : f 2 E & a d d E M,) , (here the cmtlexitg of Mz i s wed),

W e m a r k timt although for sri arbitrary ring A, M[A] always foims a foin semi-

lattiœ, the tact that in thh case M[A] f a a lattice is dependent upoa the f& that A

bas a partialaxkhg.

What fdows is a sequence of resuits nhich p&del the r a d t a of [GJ, b p t e r 21,

stated in the umtext of @-aigebras. Although the p d s that folbw are very simller to

~ f o d h [ G J I , i n s o m e c i i c r a ~ m i i s t b e m o r e c a r e h r l t h a a t b e y ~ , d u e t o t h e

fm that alfhough fm any g E A, M@) c g-(O), the taro sets need not be e@ To see

this, note that if M, ia a hyper-real ideal m A, then the totallydered f-algebra AIM,

is non-archimedian. Thus an element f E A may be f o d such that M,( f ) is Ynfinitely

d; that is, M,( f) is positive, and for e q naturai number n, A&( f) 5 l/n, (see

page 70 of [GJ1 fa the detaib of this assertion). It foiions h m the H d - J o h n s o n

representation of /, th& f (z) = û, yet z 6 M ( f ) .

A filter on the lattiœ M[A], will be called a z-Mer.

Proot a) If M, is a m&nd l-ideal d A coiddniilg 1, then for any f E I, z E M ( f )

andhence@eM[Z]. If f , g ~ I , so thatM(f),M(g) EM[A] , thenp+gZ €I,andso

M(f)nM(g) =M(f2+gZ) EM[lj. FWy, if f €1, g € A , withM@) >M(f )y then

f g e fi therefm M ( f ) UM(g) = M ( f g ) c M [ q . Thus M[l j is a Alter.

b) Let 1 = M'[A, where 3 is a z-fhr. 0 L 3, so 1 $ 1. kt f 7 g E I . Then

M ( n , M(g) E 7, and it is dear that M ( f -g) 2 M ( f ) nM(g); therefore M(f -g) E 3

and it f o l l m that f -g E 1. If f E I and g E A, then M( j g ) = M ( n (J M(g) > M ( f ) E

7. But 7 is a flter, and hem M(fg) E 7. M o r e fg ô 1, pmving that 1 is an

ideai. Now cis remarked above, M regdeci as a mepping A -. M [ A ] is a so

M[M'[7J] = 3 and hence M'[MM] = M'[M[M'[fi]J = M'[A = I . Therefore I

is a z-ideal. O

Eech member of M(A) is an f-idd, so by part (2) of (4.1.1), for each f E A,

M ( f ) = M(l f 1). It foUows thttt eanp z-idml of A is an 1-ided

An ultrafilter on the bttiœ M [A] will be called a r-uttra@ter- It is au easy C O L O ~

of the above pmpoeition that the maaâmd I-ideaIs of A and the z-ultra6Uters of A are in

onetnone cmspondm via M-imaging. hdeed the madmal 1-ide& of A are precidy

so the z-uitrafilters of A are precisely

&=M[M,I, and M, = Mt[&].

5.2 Prime t-Id& and Prime z-Filters

5.2-1 L e m : If h,g E A, z E M(h), and h(z) 2 g(z) 2 O, then z M(g)-

Roof: Lf z E M(h), ami k E A+, then (hk)(z) = 0, and (hk) (z) 2 @k)(z), hence

(gk)(z) = 0. It fonOIsg h m (4.31) that g E M,, and thefefore z E M@).

5.2.2 Theorem= Let A Oc a @ - & e h , I a z-idcol of A. ZRen the f o l l ~ n g ore equiv-

alent-

1. I is @ne,

R I containa a @e ided

9. F o + d l g , h ~ A , i f g h = O , t h e n g ~ 1, o t h ~ 1-

4. f EA, k i a a m e m k r o f M [ I ] onwhidi f dasnutciiungesign

PmoE 1. + 2. and 2.3 3. are obvious.

3. * 4. If f E A, then (f VO)(j h O) is identidy O on the dense su& f 'B of M(A),

hence (f v O)( f A 0) = O. By 2, f v O E Il or f A O E f i certainly f does not change si@

on either M ( f v O ) or M(f ho).

k a i l that if A is a commutative ring withunity, I an ideal of A, and P the collection

of ail prime ideals of A containhg 1, then nP = {f E A r E I for some n = 0,1,2, ...),

[GJ, 0.18j. Monmerl [BT, 1.51 saya that if 1 is an bideal of A disjoint from a multiplicative

system T of A, (that is T c A ïs ciosed der multipfication, 1 E T, and O 4 T), then I

is contained in a prime Gideal of A disjoint h m T.

Proofi Let I be a z-ideal d ipaigebra A, 'P, P' mpctively the dectiom of all ptime

ide& and prime &id& eonteining 1, J = n P, f = n P'. Suppose f E J. Then

for sane power of n, /" E I. But maiamal 1-ideais are prime, so M(f) = M ( f ),

and shce I is a z-ideal, f E 1. H e m 1 = J. Non œrtaidy J c f. If f J , then

T = {f" : n = 0,1,2, ...) is a multiplicative system disjoint h m I , hem f f J'.

Thedore f c J , henœ f = 1.0

ProoE Let (ID) be a cdection of z-ideais of A. Fm every P, Mt[MIIA = Ig, so

The follawing is [GJ, 2-11]; its proof is wry short and so we choose to indude it .

5.2.5 LamM: If 1 and J an ideolr in a ring A, a d d t h e ~ is cmtained in the o t k ,

then I n J is not pnpnrne.

Pro& Every 1-ideal i, contained in et least one maidmal 1-ideal. If M, M' were distinct

maximal Cideais containing prime 1-ideal P, then ~n d is not prime but is a z-ideal

containing pdme 1-ideai P, a contradiction to (5.2%). O

A Alter 3 wiil be called prime if whenever the union of two &s from M[A] belongs

to F, then at least one of them bel- to 3.

5.2.7 Theomm: 1. If P is a Mm 1-ided in A, thm M[P] is a prrrne z-Per.

2. r fF is a prime z-j&r, then M'[A is a Hme

PmoE 1. Let Q = Mt[M[P]]. Then Q is a z-ideal containhg the prime 1-ideal P,

hence Q is prime. Maeover, M[Q] = M[P]. Suppose M ( f ) U M ( g ) E M[P]. Then

M(fg ) E M[Q], hence fg E Q, thetefore f c Q, or g E Q. Thdore M(f) E M[Q] =

MP1, or M g ) E M[Ql = M m - 2. Suppose 7 is a prime 2-Hter and fg E M'[A. Then M(J)UM(g) = M(jg ) E

M[M'[A] = 7, so M( f) E 7, or M(g) E 7. Tkefore f E M'[A, or g E M'[fi,

thus M'[fi is a prime z-ideal of A. O

5.3 N, and Prime z-ideals

In [HJj, the Gidd

is infroduced. ExpteSged in our crnn notation, we ckPn that

To see this suppose that f E N,. Then a ndghborhood U of x may be fonnd on which f

vanïshes. If g E A, then R ( g ) = g'[Bj is a dense open subset of M(A). Thus R(g) n U

is a dense subset of U an whïch fg vienishes; it fdows thst (fg)[w = {O). As g aas

chosai arbitrsrily, by (4.22), for erery y E U, j E Mg. Thus the open neîghbrhood

U ofx is containai in M ( f ) , showhg that N, C {f E A :z E intM(f)). The reverse

containment is an immediate CoIlSeQuenœ of (4.2.2).

It folIows easily fkom the secand part of the HenrikseaIIohnson Representation The-

aem that n M[N=] = {z}, end so NI is contaïneci in the unique maximai Cideai M,.

PmoE 1. ts *. '

2. If f E N,, tben z E intM( f ) . By the H m - J o h n s o n representation theorem,

there is a g E A, O g 5 1 with g[M(A)\intM(f)] = O and g(2) = 1. Then g M= and f g = O. C 0 n . y suppase that g t M, and jg = O. Then z $ M(g), yet

M ( g ) U M ( f ) = M(fg) = M(A). Ttberefore M(A)\M(g) is an open set containing z

that is contained in M(f), heme z E intM(f). Therefi f EN,. O

5.3.2 Theoram: An 1-ided 1 of A is matuined in a unàque mmhnd 1 - M M, if and

o @ i f I 3 & .

The foUowhg comUary is [HJ, 2.10], where the result is obtained nithout the use of

z-ideals. Nmheless, the above illustrates that the notion of 'z-iddl is usefd in the

conta of 4algebras, just as z-ideals are useN In studying C(X).

5.3.3 Comhry Let P & a -e l-ideai of * - & e h A. Then them is a unique

x E M(A) auch that N, C P C MI, and Nz i9 the intmection of dl pime 1-ideais

cmtaining it.

PmoE N, is a x-idd, hence the last statement follows. It has already been s h m that

every prime i-ideal is cuntaüd in a unique maximal I-ideal, so by the above theorern the

d t f d m * 0

5.4 An Application: P-algebras

A *-algebra A will be d e d a P -dg* if every prime t-ideal of A is a maximal bideal.

By the deaaition of a P-spgce, [GJ, B J j , C(X) is a P-aigebra if and ooly if X is a P-space*

5.4.1 T h m m nit foUowing a equivalent for a 9 - & e h A.

1. A i& a P - & e h

2. Eumg z-idcd k an inkrsedt-ML of mazimal 1-ideab.

3. Fm & z E M(A), N, = M+.

4. Fmeach f E A, M ( f ) is open

PmoE 1. 2. Every z-id4 is an intersection of prime 1-id&.

2. -+ 3. N, is a z-ideal cmtained in p&y one maximal 1-ideai, nrunely M.. It fdm

brn 2. that N, = M;.

3. * 4. Let f E A. If M( f ) = 0, then M (f) is open, o t h d e take z B M(/). Then

f E M, = N,, whence z E int(M(f)). M o s e M ( f ) = int(M(f)).

4.*3. M , = { ~ E A : z E M ( ~ ) ) = { ~ ~ A : z ~ i n t M ( f ) ) = N , .

3. 1. If P ïs a prime t-ideal of A, then there is aa z E M(A) such that N, c P c Mz ; by 30, P = Mz. 0

C o m b ' i nith (4.3.3), the above yidds the foUOfRing imrnediate coroilary., a d t

which in view of (5.43) is just [gT, 3-10] in disguise.

The rest d the chapter is devded to pmving (5.4.3) which characterizes uniformly

dosed P-algebras, and which, far the sake d the diseussion that foliows, ne presently

state. Recall that an algebra A is called regulw if for every f E A there is a g E A such

that / = f g f . (5.4.3) s h d d be corn& with [G J, L4.%], which is the theorem that it

generaüzes.

5.4.3 Theotem If A is a uns'fdly closed @ - * e h , then the folloun'ng are equivdent.

(1) A is a P - @ e h

(2) Fbr every 2 E M(A), Nf = M,.

(3) For criery / E A, M ( f ) is open

(4) Eaery t-idcal is O z-idcol and eoety l - p t t P t t m i ~ i d d b prinei'pd.

(5) Eu- idd of A is a z- ïded.

(6) F w en- f, g E A, the ideal( f, g) is the pnpnnd@d ideal ( f + g2).

(7) A is a n01Jor @-dgebm

(8) E a q prime i&al in A i s a mazimal idecrl.

Evidently, am- d d y c l d *algebm, the P-algebm are precssely the reg-

uiar Wgebras, and as & - examples abound- As mentioned earlïer, C(X),

where X is a P-space, ïs one such example- M e r s include the Baire functions on R and

the Lebesgue meamrable furilctions on R, each of which if desired may be taken modulo

its ideai of functions that vanish almat ewrywhere, m, 3.101; yet aiiother example of

a UIUformly ciosed regular Palgebra is the epimorphic huil of C(X) which is exaniined

in the pre-print m, by Raphael and Woods. B. Brainerd in the late fifties studied

reguiat F-rings, (F-rings lue Wiifody c l d *algebr., see se] for rdrefeces), and, as

they pertaîn to z-id&, reguIar rings were studied by Mason in [W. Considering then

the attention that regular Palgebras have ieceived over the years, it is not surpnsing,

(though admittedly the author who is rather was quite smprised), that (5.4.3) is

not eritirely new. W e do M e v e havever that the impLication (1) * (7) is new, and it is

for this proof that the following lemmas are requiréd. W e comment that it would be in-

teresting to how whaher (1) * (7) hoIds if it is n a assumeci that A is d o r d y c l d ,

(the author thinks not, although (7) * (1) does still hold). With no fbrther cornmentary

we mw p m the t h e m .

As noted in section 5.1, for g E A, M@) C g'(O), but in g e n d eqdw does

not W, a phenornenon which can cause problems. The next 1emma shows that in a

uniformly d d P-algebra this probkm may be corrected.

5.4.4 bmma Let A be a unifownly do& P-algeorcl. For any g E A with g 2 0, t h m

ic a unit k E A with k 2 1, und gk > 1 on M(A)\M(g) sudi that M(g) = M(kg) =

(kg)'(O). Minwuer, ÿ f E A is sudi th& M ( f ) = f'(O), then Uien ic an e E A a<&

thut e f = If l*

PmoE By (5.4.1) M (g) is open, so F = M(A)\M(g) is a compact subset of M (A). Let

z E F. Then z M(g), so g f M,. Thdore there is a k, E A such tbat (O&)(%) > Q

without l a s of generality, k+ 1 1 and (g&)(x) > 1. For each z E F, choose such a kr

and d e 5 e the open su- U(z) d M(A) to be (y E M(A) : &)(y) > 1). Then for

everyzE F, z ~t((z),soby~ornpactnessdF, zi, ..., z,auibeebocwifiomFm&that

F C ~ & U ( ~ ; - ) . Let k=k,+ ... CL. T h e n k k l (henceM(k) =@), d g k z g k ,

for 1 I i I n. If y E F, then y EU(Z:) fat m e i, so @&)(y) > 1, therefbm (gk)(y) > 1. Hence (gk) (y) > 1 for ail y E F = M(A)\M@). Thus

Therefore M (g) = M (gk) = @k)'(O).

To prove the last statement in the lemma, if f E A is such that M ( f ) = f '(O), then

def iwe:M(A) -Rby

Then esch elVi is continuous, and M(A) is the wdnt union of the open sets Yi, whence

e is cûntinuous. It is dear that ef = 1 f 1. O

5.4*5 bmma Let A k a u n i f d g cloaed P - @ e h rinth f,g E A, g 2 O. If M@) C M ( f ) , € h e n k i s u n h ~ A a r c h t h a t h g = f.

Proofi By (5.44, we may nithout loss of p e d i t y assume that M(g) = gh(0), and

gl[M(A)\M(g)] 2 1. Using (4.3.2), we may take f* = (1 + I / I ) - ' f . Then -1 < r I 1 and since f œ is a unit multiple d /, p and f are eonteuied in precisely the same set of

xwâmdl-id&, henceM(r) =MM( Thdoreg+(O) = M(g) C M ( f 3 C (/*)'(O)-

Desne k : M(A) 6 y R by

[Here fa is taken to be O if g(z) = oo. This is welldehed since f' 19 bouiaded] Then

LI [M(A)\M(I)] is conthmous, as is k[M@) = O. S i M(g) ïs open, it follm that h

is itselfcontinuous. L k t , becauseg 2 1 an M(A)\M(g), it foUows that -1 I k I 1, and so k E C(M(A)) = A* c A. A h , for ewry z E R(g) , (the r d part of g),

k(z)g(z) = f*(z). But kg E A and Rb) = %kg) is dense in M(A) hence kg = f*.

Themfiore [Cl+ If l)k]g = f - O

Proof of (5.4.3): The equivaence of (l), (2), and (3) rn estabkhed in (5-4.1).

(1) * (4) Let 1 be an 1-ideal of A and suppose that f E A, g E 1, andM(f) > M(g).

Then lgl E I , and M(1gI) = M@) C M ( j ) . By (5.4.5) there ïs an h E A such that

hlgl = f , therefi f E I . Hence I is a z-idd of A.

N m let f E A and consider (1) and (nt = {s E A : :SI [fgl for some g E A).

Ceaainly (f) c (f)i. Suppase then that s E (f)c. Take g E A such that Isl< lfgl. Take

k E A audi that k 2 1 and M(lfg1) = M(klfg[) = (kl fg()'(O), (ushg 5.4.4). k 2 1

so k[fg[ 1 jgl 2 181, and therefore M(k(fg() C M(ls1) = M(s). By (5.4-5) there is an

h E A such thst hkljgl = s. Non klfg( = Ikfgl and M(kfg) = (kfg)-(O), so again

by (5.4.4), thae is an e E A such that ekfg = [kfg[ = klfgl. Thus hek fg = s, that is

(hekg) f = s, hence s E (f ). Thus ( f ) l = (/), prwing tbat every 1-principal ideal in A is

principal-

(4) * (5) Let I be an ideal of A, and suppaae f E 1, g E A with (g[ s [/[- By (4)9 (f)i = ( f ) c 1. But g E ( f )~ , so g E 1, shoning that I is an 1-ideal.

(5) * (6) Let f , g E A, and consider the ideah (1, g), (fL + g2). Certainly (f, g) > (f2+g2)- Now M(fZ+g2) = M ( f ) n ~ ( d =M(f)nM(g) , the iatterset amtaïneci

inbothM(f),M(g). But f 2 + g 2 ~ ( f 2 + g 2 ) , a n d ( f 2 + # ) isatideal by hypothesis,

h e m I,g e (f2+g2). Therefore (f,g) c (f2+g2)-

(6) * (7) Let f E A. T8king g = O in stataeemt (6)l ( f Z ) = (f) . Therefore f E (f2),

h e n c e t b i s a g ~ Asuchthat f =gf2. Thus Aisregular.

(7) * (1) Let P be a prime 1-ideal of A, and suppose that M is an 1-ideal d A that

properly amtains P. Take then j E M\P, and let g E A be such that f = fag. Then

f - f2g = O, so that f (1 - fg) belongs to the prime ideal Pl henœ 1 - j g E P C Ml

aud fg E M. Téerefore 1 = (1 - fg ) + f g E M, and M must equal A, proving that P is

TImdmal.

This establishes the equivaience of (1) through (7). Since every 2-ideai is an 1-ideal,

that (1) (and (5)) ïmply (8) is clear; the c o n . , (8) implies (l), Is obvious. O

Chapter 6

Closed ideals of a-Algebras

C l d 1-ideah of 9-aigebras have been studied in [Eil by M. Heariksai and p] by D. Plank. Nice characterizaticms of ci& 1-ide& were attained in each of the doremen-

tioned papers, but these descriptions are neither aIgebraic nor intemal to the bideais

themselves. In this ehapter ne endeavour to Bnd mch a characterization of the clad

1-ideah of a a-algebra uaing the concepts of strong divisib'ity and Ad&. Admittedly

the attempt is not entirely suaxd& hwver in section tno both necessary and sufncient

conditions on an 1-ideai are f d to ensure that it is closed. Moremer, it is shown that

under certain canstraints placeci upon a +slgebra, these two conditions agree? and hence

an algebraic charactetbation of c l d 1-ideais is attained.

W e kgin by ggeeralizing the results of chapter t h t section one, coaceraing prime

ideah and closure in C ( X ) to arbitrary ieaIgebras.

6.1 Closed Ideais and Prime Ideals

Let A be a Wgiebra and kt M(A) denote the miudmal ided spaœ of A. Recall h m

section 6, that f a every z E M(A) the set

is an 2-ideal d A contained in -tIy one maidmal 1-ideal, namely M,.

PmoE Let z E M(A). Since N, c MI, K c To pmve the reverse containment

let a Q M, and let E > O. Then a(=) = O and since a : M(A) -. y R is mntinuous,

U = a+[(-E,E)] iBanopenneighborhoodofzQM(A). Let b = [ ( a - E ) V O ] + [ ( ~ + E ) A O ~ .

Then b E A, Ib - al 5 h and q(lj = {O). Heiiee b E N,, and so a E K. Thdore

M, CZ, w h e n c e & I ; ~ x - O

Non as sh<w in section 6, if P is a prime ideai t-ideal of A, then there is a unique

z E M(A) such that N, c P c M,. By the pmopoeition = P = x, fkom which it

foliows that nonomeidmal prime 1-ide& of A are never d d Suppose however that A

is a unSody c l a d 4algebra Then by (P, 2-61 the c l c d rnadmal 1-ideals of A are

precîsely the real Gide& of A. Thus if M, is a real 1-ideal of A with P C M,, then

~ = P = M , . Hacewehave

8.1.2 ComlLry Let A be a unif- closed iI?-algebm.

1. IfM, r( a d l - i d c d ~ f A , M ~ . 2. r f P isapMne1-ÙiedofA, ~ 1 c n ~ i s a ( n e c e s s c i r i l ~ n u u 5 d ) I - i d c a l i f and onlyif

the uniqrie tniuimoll-ided M containing P is nd; in th& aase P = M .

6.2 z-Ideals, 2-ideals, and Strong Divisibiiity

At this point we attempt to -the ckac&&ation (3.i.4) of cloeed id& of C(X)

to various types of @-algebms. To begin are consider amther extension d the notion of

z-ideal in the contact of a halgebra

Let A be a @-algebra. Let R(A) denote the spece of teal maximal Cideals d A, and

for each a E A, kt S(a) = (M E R(A) : a E M). As More, let M(a) = {M E M(A) :

a E M). For the sake of compatison, we restate the definition of z-ideal.

6.2.1 Deiinition k d A A a 4L&e&ru, a d k t I üe on i d d ofA.

1. I EOÜI h d k d a z-ided if

(M(b) 3 M(a) and a E I) * b E 1.

2. I wiil be d l e d a 2-ideol if (S(b) 3 $(a) and a E I) =, b E 1.

Clearly then, every 2-idd is a z-ideaL It is a d-known result that if X is a

topologid space, then the above deanition of z-ided agreeo with the usual notion of a

z-ideal in C(X), [GJ, 4. In fm:

6.2.2 Proposition If X is a topo^ spaee, and I ir an idml of C(X), then I is a

z- idd if and d g if I is a Z-ùkd

Proofi Suppoee I is a z-ideal of C(X) and suppaw S(g) > S( f), f E 1. Let

p E Z(f)- Then f E M'p and M, Ie d. Thedore g E M,, whence p E Z(g). Thdore

Z(f) c Z(g), and Z(f) E Z[q, Z[q a z-flter. It fdows that Z@) E 2[1]. But I is a

z-ideai in C(X) and t h d m g E 1, sbowing tbat I is a Z-idd. O

Indeed in (6*2.q, we nill shoa that if A is any ipalgebra of d-valued ninctions

thst is dased under i n d o n , (definitions of these concepts appear on page 50), then the

z-ide& of A ere 2-id&- (6.2.9) provides en exampIe of a *dgebra in which *id&

are not necessarily 2-id&.

To pmve the next few theotem, the fo- dennttims and results h m the papers

by M. Hentiksen ami F] by D. Plank are n d d

ln [Hj an idtol set is defined to be a dosed subset A d the space M(A) of maximal

1-ide& of hilgebra A such that whenever a E A and a[A] = O, then (&)[A] = O fm d

b E A. It is shown that

[H, 2-41 An 1-ideal I of a unifi~rmly c i d -bra A is c l d if and ody if there is

an ideal set A of M(A) such that I = {a E A : o[A] = 0).

[P, 2.61 If A is a d d y dosed @4gebra, then a maximai 1-ideal of A is c i d if

and d y if it is reai.

[P, 3.1 If I is either a C i d or maximal ideai in the uniformly cbed 4Lalgebra A,

then I is an 1-ideai.

Recall also Erom chapter fm, that if A is unifody c l a d then its sub4Wgebra

of bounded elements A* is L-isomorphic to C(M(A)), and as such C(M(A)) may be

regarded as a mb@-aigebra of A.

These d t s are used in the pmofs that fdm.

P m E Let I be a d d ideal of d o n n l y ci& edgebra A. Let r = I n A*. Then

I* is a closed ideal of A* = C(M(A)), and therefore is strongly divisible by (3-1.4). Let

(h) be a countable subset d 1. A is doaed under bounded Inversion, so for each n,

(l+lt~,J)-~ SrUrtsinA. B ~ t e a c h ~ ~ ~ e A * n I = r , sobythesgoogdïvisibili~of

P thereïs ab E P , and (a) CA*sachthatfoieachn, b - R . = c e i . Beoceb € 1

and for each n, b 4. - (1 + [a,,[) = on, proving that I is s-y divisible- As a d d

ideal d uniformiy c l d A, 1 is an 1-ideal, p, 3$, so let A be an ideal set such that

I = {a eA::a[A] =O}. SupposeM(b) >M(a) a n d a ~ I. IEz~Athenfora l l c~ A,

(ac)(z) = O, hare a E M+, by (42.2). Thetefixe 6 E Mt, hence b(z) = O. Thus Q[A] = O

and thdore b E 1. This proves that I is a z-ideal. O

6.2.4 Thwrem Any stmngly divisible 2-ideal of a O s l g e h A is &sed in A.

ProoC: Let I be a strongiy divisible 2-ideai of 9-dgebra AT let b E Î. T& (b.) 8 coutable sukt of 1 such that for each n E N, Ib - 41 < l/n. Then by &e strong

d iv i s ib&~dI , the~e icrsnodIand(~) ~Asuchthetfcaeachn,a-an=&. Suppae

a E M,, where M, ie a mai ideal of A. Then a(z) = O end t h d m for each n E N,

u -u,,(z) = O. Thereiore for ail n, l /n > la -u,,(z) - b(s)l = Ib(z)l, henœ lb(z)l = O;

thedoré b E M, by (4.2.3). This shows th& S(u) C S(b). But a E I and 1 is a Z-ideal,

S O ~ E I . Therefm I=?,henœ~ischsed. O

A m t d question to ask is whether or not stmngly divisible z-ideh in a-algebras

are always ciosed. Certainly an argument similar to the one given abme wiii not answer

the question in the nnimiative; hol~ever, if we assume that A is a P-algebra, then the full

converse to (62.3) holds. P-*bras were characterized in the fourth section of chapter

five.

P d Let- I be a strongly divisible z-Meal in A, chocse b E 1. As in the proof of

(6.2.4), we may h d a E I and (h) C A such that for each n E N, la - bl < i/n.

T w x E M(a), for etxù n, (a-&)(z) = O, hence b(z) = 0, and thedore M(a) c b+(O).

But A is e P-aIgebta, so by (5.5.1), M(a) is open, hence M(a) c int(b'(O)), and since

int(b'(0)) c M(b), ne have M(a) c M(b). p e note that were A mt sssuned to be

a P-bra, M(a) C M(b) couki not be shmm hi this mruiner]. But I is a z-ideal in A

a n d a ~ I , b e a Q b € I , p m v i n g t h a t ~ = T . O

R e d b m that a @-a@bra A is d e d an & e h of naCvduedfinctions if its

space of real mai<imal ide& R(A) is deme in M(A); in th% case A can be embedded

as a sub*algebra of C(R(A)). A 4baIgebra of reai-dued functions A is dus& un&

inildon if, for dl a E A, a'-(O) i)R(A) = 0 implies (a)i = A, where ( u ) ~ denotes the

d e s t t-ideai in A containing a. The folowïng is p, 4-61.

6.2.6 Q A à& a @-algebm of 4-ual tred findim tuhich is closed under inaemion, then

for each z E M(A),

M, = {a E A : z E (~'(0) ~ ' R ( A ) ) ' ) .

We use this thecnem to show that witbin the class of -bras of real-valud func-

tioas, that every z-ideal is a 2 - i d d the property CM under inversion.

6.2.7 Theorem Let A k a 4D-&ebra of d - u a l u e d jhcths. W h tk folIowing are

equiuaknt.

(1) A is dosed vnder inversio~

(2) z-idcab in A on 2-üièais.

Proof (1) =+ (2) Suppose I is e z-ideal and S(b) > S(a), a E I . Let z E M(A)

and suppose a E Me. Then by the above, z E (aw(0) nM(A))- . Non suppose

y E c'-(O) n R(A). Then a(y) = O and y E R(A), so a E MM- But S(b) > S(a) , so b E M,, hencs Ky) = O. T h d o r e y E bh(0) nîZ(A), whence (ot(0) i)'R(A))- c @-(O) n R(A))-. By the above th- it fdows that b E M,, h e ~ z M(b) > M (a).

But a E 1 and I is a z-ideal, so 6 E I . This shows that I is a 2 - i d d

(2) =+ (1) Suppoaïng A is not c h d mder inversion, ne may take a E A such that

ak(O) nWA) = 0, yet (a)l # A. Tben S(a) = 0, and a madmell-ideai M+ may be found

such that c Ms. Thus if b E A\M,then S(&) 3 >(a), a E M,, yet b f M,. It fdows

that the z-ideai M, is not a 2 - i d d O

As a conqgenœ d the above we get the fou- characterization af the clmeci

ide& in taR, b m d elecoes of r3,aigbm.

An example of H&n and Jchmon [&T, 4-51, does the job. Let A = {/ E C(R+) :

iim,- f (z)e- = O for di r d a > O}. Then A is a unjfordy chsed bigebc8 of real-

valued functions. A* and C(R+) are isomorphic, so M(A) = M(A9) = M(F(Rf)) =

m+. Take g(z) = eW*, (z E R+). Clear1y then g E A. M o m , g+@) = m+\R+ =

M(A)\R(A), thereforel M (g) c M(A)\R(A), and henœ S(g) = 0. But l/g g! A, so

(g) # A, and thedore there is a maximel 1-ideai M cwtaining g. M is then a tideal

but is not a 2-ideal. For take any h # M. Then S(h) > S(g), g E M, but h $ M .

The Hexuikn-Johnsoa repre~entation theorem has been used to characterize C(X) ,

for various homenmoiphipm desses d speces X , aigebraically nithin the class of a- algebras. Cbaracterizations of sewral such elasses of (rdcompact) Tychonoff spaces

may be f o d in LHJI and many other pepers. N e e l e s s a 1- open problem

concerning @aIgebms a&, %ml an intanal &te&ation of C(X) for X a Tychonoff

spaee within the dass d -bras', [&, prob1em 51. h the temainder of this section a

discussi~~ and modification of a hdgebra characterbtion of C(X), for a Lindelof space

X, due to D. P W , [P, 3.7, WU be presented. We remark that Plank's result receives

high praise in each of the mwey articles m2] md [BI by M. Henriksen. F i y ne p m

pose by way of conjecture a 4Lalgebra chatacterization of C(X), for X a weakly Lindelof

space-

In W, P M cab a CaJgebra A normal if every dosed 1-ideal of A is contained in

a c l d msximal bideal of A, and pmves that if A is normai, then R(A) is a dense

Lindelaf subapace of M (A), (and heme A is a Mgebra of real-vaiued functions). He

4 s A clmeci under inYeTSion if (a)< = A whenever a E A, o is not a zero-diivisor in A,

and a 6 M for all M f =(A). This concept agrees with the definition of closed under

inversion given earlier providecl that A is a @-aige!bra of real-valued fwictions. Since we

shaU only be concenieci with this property in the ccmtext of QLeigebm of d - d u e d

tunctions, we wii l not distfnguiah between the two dehitions. P W p m ~ s the foRowing

theorem, which is itseif a modification of a theoileai fomd in the HenrikSen-John p a m

M. Note that A is not 8ssumed to be a @-aigebra of real-valued functions; it fo~ows

from the aspinnption that A is nomal, [P, 3.11.

6.2.10 [P, 3-11 Theore!m= A non-tria @ - & e h A is 2-LPomofpIiic to C(X), fm some

Linde1of gpace X, if and only if (i) A is u n i f d y ciosed,

(ii) A is closed under inversion, a d

(Ci) A is normal.

Using the voc8buky of the present section, we! pprovide the following refonnulation

of Plank's d t , which phrased in terms of strongiy divisible, rather than ciosed ideals,

perhaps seem slightly more aigebraic in character.

The bracketed t is meant to indiate that it is optional.

Proof Neeessiw fs dear. C o n d y assume coaditions (i), (ü), (iii) and let I be a

d d ideal of A. By (69.3) 3) is a -y divisible z-ideal in A, t-ore containeti

in a stmngly divisible maximal 1-idd of A by ci). But by (ii) M is a stnmgly divisible

2-ideai, and therefore c l 4 by (6.2.4), showhg that A is no& Non by P W s d t

(quoted earfier) A is then a @-aigebra of rd-valued fiuictims, hence by (3) and (6.2.7),

A is c l d under inversion, and sufficiency foilm h m (6.2.10). 9

To conclude this section we O& the following conjecturey which despite persistent

sorts ne weze unable to p m ~ Due to the faa thst sny C(X) is (0-isomcxphic to C(vX),

we may as well assume that X is realcornpact. Recall that a strongiy divisible ideal

comprised M y of d v i s o t s is d e d neighborhood strongly divisible, or simply

IiSd-

6.2.12 Coqjectuie A non-tri&hZ O-aigebm A ia Glsornorplric b C(X) for s m e weaklg

LindeIof (dcompact] s p a ~ X if and onlu if (i) A is u n i f d g closed,

(ii) eony z-&hl in A is a 2-ideal, und

(ui) eu- nad idcd in A is amtained in a stronglg divisible maximal 1-idcal of A.

We note by way of evideece to support the validity of this statement, Theorem 3.3.4,

which under the addftfond hypothesis of re81compactness of X quivalently reads:

X is weakiy Lincielof if and only if evay nsd ideai of C(X) is containai in a stmagly

divisible maXima ideai of C(X).

Hence the amditions (i), (ii), and (a) of (6.2.12) are certainiy neœwuy We alpo note

that a r g u i . as in the proof of [P, 3-11, it folloas b m (i), ci) and (fi) that A is a 4

dgebra of rd-vslued fiurtions; Ullfktuaately it seans that a p m f of (6.2.12) would at

this point re~uite 8n dternative to Plank% methods.

In this section an attempt is made to describe the relationship between an element of a

@-algebw and its correspondhg mset in a -or aigebra with respect ln a given l-ided-

It is weii-knm that if M is a maximal icieal of C(X), then M is red if and aoly if

Z[Ml is cheed unda corntable intersecticm. The motivation for the fdowing wss to

obtain such a chrrracterïzation of the nal maximal id& of a given *aIgebra, in ternis

of the subeets M( f ) , (f E A). UnforGimately, auch a nice result was not, in the general

case obtained, though we did &tain some partiai d t s . We develop the foltowing t~ in

cbapter 5 of [GJj.

Recall h chapter four, that if I is an Cideal of A, aad f E A, then I ( f ) denotes the

image of f under the crinnnical (E)homomorphism h m A ont0 A/ I . More reading the

proof of the next statement the reader msy wish to review 1.1 of h p t e r four, and the

discussion of the HenrikSen-Johnsan representatim of an dement of a given kdgebra,

which is f d in the same chapter.

6.3.1 Prqnmition: Let A bc a + - @ e h .

1. y1 ia a z-idcUl in A, Uhen I(f) 1 0 Band onlu if f i9 non-ncgatiae on sorne m*noer

of MM- 2. rf f is pogitive on s o m mnnber of MM, nihm I t a z-ideal of A, then I ( / ) > 0.

ConverSejy suppose that g E I and f (z) 2 0,for evay z t M@). Then f a each

E M(g), &(f) 2 0, heiree by (4.1.1), f - If1 E M& Thus for eseh z E M@),

z E M(f - [fl) - that is M@) c M(f - 1 11). But I is a z-ided, and g E 1, therefbre

f - 1 f 1 E I, and h e m I(f) 2 O.

6.3.2 Corn- I f z E M(A), Uicn f (2) > O if and only if f is noicnegutive on a

m e m k of M[M&

Pm00 f (2) 2 O if and only if M+(f) 2 O, by the HenriksenJohnson representation

theorem. The result follm b m part 1. above. 0

As a second caollary ne get the foUowing exteosion of Thmf1ll5.2.2.

6.3.3 Comllary: If1 i s a z-idead, then AIT is totallu tmkred if and only i/ I b pBme.

PmoE A/I is totaily ordered if sod ody if for every f E A, I ( n 2 O or I( j) 5 O, if and

only if for every f E A, f( f) 2 O or I(- f) 2 O, if and only if for every f E A, f does not

change dgn on a member of M [ q , (6.3.1), if and oniy if I is prime, by (5.22)- 0

In fact aescmiing d y that P is a prime 1-ideal, not n d y a z-idd we can show

that A/P is totdy oràered. For take f E A. Cleariy, ta every z E R(f) = f+(Rj,

(f - IfI)(f + If I)(d = o. But W f ) is &me h M ( 4 , so (f - If l)(f + IjI) = 0. P is prime so f - 1 f 1 E P, or f + 1 f 1 E P, hence P(f) = P(l/ l) , or P ( f ) = P(-1 fl) for every

f G A . Thus P ( f ) LO,ocP(f) SOforevery f P : A - A/Pisdetpreserving,

(4-1-l)), hence A/P is totdy ordered.

Note that since maximal ideais needntt be t-ideais, w, it ïs also tnie that prime ide&

needdt be 1-ideab, unüke in a C(X), [GJ, 5.51.

6.3.4 Lernma: Let A be a O-dgebtU, f E A, and kt X be a posüà~fe ma2 numk. Let

Tn«i taking g~ = X - (1 f 1 A), fi c M@A) c FA.

Proof: The equivalence of 1. and 2. is clear from the Henriksen-John representation

of f * 2. * 3- M[M,J = {M@) : z E M(g)). ClerY1.y then, if 1 f (z)l= m, then f is unboded

on every manber of M[M=]. Conversely suppose that 1 f(z)l = X < oo and take U =

r [ A - 1,A + 11. EU = M(A), t h f Ie bomded, henœ 3. is false- ûtherwise, U is

a dosed neighborhood of z, and U # M(A). Ushig the norxnaiity of M(A), take V a

c l d neigbborbood of z sueh that V c in&(. Then ch- g E A such that gM = O,

g[M (A)\intL(I = 1. Then z E intg'(0) C M(g) c 24. Hence M(g) E M[M,], and since

6.3.6 Pmpdtion: Stqpose tlht A is a u n i f d y &ad baigebnz, and z E =(A) - lhen for any (f,,) C Ms tlien is an s E Mz such Bat M(s) C &M( fn). In o h

aionis, mery aountabie subfami(y of M[M,J hos a laver bard in M[M,].

ProoP: For each n, M(fn) = M(&), so without loas of generality we may ansume

that O 5 f, 1, for every n. Take for each n, = CLl 2-'fk. Then (&) is a

Cauchy sequence in u n i f d y ci& A, so there is an s E A Juch that - S. As a

real ideal of Uilformly c l d A, M, is cl& in A. Eeeh E M,, therefore s E M,.

Also because eaeh 4i 5 s, M(s) C M ( h ) , thdore M(s) c h M ( b ) . But for each n,

M(&) = M(fk), h m whieh the pmpositian follm. O

In the context of W U f d y cl& regular @-aigebras, the next hposition generalizes

(3. 1.2) .

6-3.7 Proposition Let A A a a-&eh 1 an 1-ided of A.

1. r f A is u n i f d g close4 regular and eoery eountabk subfamily of M[q hus a loluer

h n d in M [q, then 1 4 atnmgiy divisible.

2. If I is stronglg divisible, then mery coz~ntabk aibfamily of MM has a bwer band in

MM-

P m E 1. Let (fn) c 1. By asamiptiai there ie a g E I such that M@) c M( fn) for

each n. By (5.4.5) each is a multiple of g, hence 1 is sfr~ngiy divisible.

2. Let (1,) C I so that (M( f*)) is a corntable subfamily of M [ q . Choming g E 1

such that g divides each fn, it is dear that M@) is a member of M [ a containeci in each

M(fn)- O

P d The equivaience of (2) and (3) is immediate h m the above pmposition.

(1) u (2) is (63.8)- O

Question: 1s M[A] = {M( f) : f E A) c l d under anintable intersection as is true

with the lattice d zerwets on a topologid spsce ? Can it at least be shown that M [Al

is closed d e r cornitable intersection under the additional hypotheses that A is regular

and/= dormly c l a d ?

RecaU that a -bra is normal if every cbsed Lidd of A is contsieed in a closed

maximal I-idd.

Pro06 1. O 2. is b u d i a t e from (6.2.8).

2. 3. If I is a strongly divisible ideal, then by (6.3.3, J = M'[M[q] is a strongly

divisible z-ideal antainhg I, by 2. then, I is containeci in a stmolgly divisible maximal

ideal.

3. + 2. is obvious. 0

With the above in miid, the folhhg the~rem, which was h t dhmmed by B.

Brainemi, and later modifled as stated bdow by D. Pl& in [Pj, i9 in f a a purely

algebraic &mcte&&ioo d C(X), whoe X is a LùdeW P-spaœ. Of CO-, amdition

2, can be repIriced with any of its equivaIa fams ris found hi (5.4-3).

A @-algebra is calleci ueamplete if evay amtable subeet of A that is baunded above ni

A bas a supremum in A. A a m p l e t e W b r a is necessarily uni fdy c l d , M.

6.3.10 p, 4-21 Theoram: A twn-trirricJ a d e h A is isomorphie to C(X) foc some

LindeEf P-spaœ X if and o n l ~ if 1. A is a-cornplde,

2. A is regular, and

3.Aisnomiol

Bibliography

[Al] F. A z 8 t p ~ & , Algehic poperties of some compact sp~ces, to appear.

PP] G. Bioff and RS. Pierce, t a t t i C e - o M rings, An. A d . Brasil Ci., vol. 28

(1956), 4249.

[GJI L. Gillman and M. Jerison, Rings of Contsnuour finctMns, Springer-VerlagT 1976.

[Hij M. Heniiksen, U n i f d g cloaed idcds of u n i f d g t c e d @ e h of eztenàeù nBl-

d w d f i e = , Sympoaia Math. 17 (1976), 49-53.

[Hq M. Henriksen, Rings of continww findions jhm an algebmic point of view, Or-

demi Algebraic StructuresT Kluaer Academic Publisbas 1989, 144474.

M. Hemiksen, A mmq of &rings a d smne of heir g e d k i z t i m , Ordered Alge-

brdc Structures, K1uwer ACBdemic Publishclrs 1991,l-26.

[ai] M. Hentiksen aiid DG. Johnsoii, On the skudun of a darr of Archimedian lattiœ-

ordend & e h , Fùnd. Math., 50 (1961), 1394.

J. Porter and RG. Woods, J!BtEamrions and AbsoI'~1t.e~ of Hausdarff S', Springer-

Veriag, New York 198%

R.M. R e p M d RG. W h , Tihc e p i ~ h i c huü of C(X), to apperu.