Semigroup Forum Vol. 46 (1993) 109-115 �9 1993 Springer-Verlag New York Inc.

R E S E A R C H A R T I C L E

Product Integrals in strongly differentiable power associative groupoids

M i t c h A n d e r s o n

Communicated by Jimmie Lawson

The existence of product integrals in differentiable semigroups with neighborhood of 1 homeomorphic to a Banach space and with multiplication strongly differentiable at (1, 1) was shown by Garrett Birkhoff in 1938. This result was then used to transform the multiplication into canonical parameters. We show here a sufficient condition to guarantee the existence of product inte- grals in differentiable power associative groupoids with mult ipl icat ion strongly differentiable at (1, 1).

Suppose X is a Banach space, U is a subset of X containing 0, and V : U x U -* X satisfies V(x,O) = V(0, x) = x for each x E U. If n is a positive integer greater than 1, denote V(x , V ( x , . . . , V (x , x ) . . . ) ) , the product of n-xrs , by x n whenever the product exists. Define x ~ = 0 and x 1 = x. The statement that V is power associative means V(Xn,X m) = X n+m whenever each of n and m is a nonnegative integer and the product exists. If y E D, denote left t ranslat ion by y as L~; that is, L~ : D --~ X by L[(x ) = V ( y , x ) . Also, let L~(z) denote Y(y , L ~ - I ( z ) ) whenever n is a positive integer greater than 1 and the product exists.

Suppose U is an admissible subset (U C U ~ of X and V : U x U ~ X is a power associative multiplication defined on U which is strongly differentiable (see below) at (0, 0). Moreover, suppose V satisfies the following condition:

(C1) there exist positive numbers d and M such that [LJ(x) - LJ(z)l < M i x - z I for each j e { 0 , 1 , . . . , n - 1} whenever each of x, z, and y is in U and each of x, z, and ny is within d of 0.

If f : [0,1] --~ U is strongly differentiable at 0 and f(0) = 0 then there is a t n

s > 0 such that if 0 < t < s then j i m ( f (~-) ) exists. In other words, (C1) is

sufficient to guarantee the existence of product integrals.

R e m a r k . It should be noted that in the associative case condition (C1) is a trivial consequence of the strong differentiability of V at (0, 0), where M = 2.

Before proceeding to a proof of the main results we indicate some back- ground.

A is an admissible subset of the Banach space, X , means that every point in A is a limit point of the interior of A, A ~ Let f be a function with domain the subset D of the Banach space X and codomain contained in the Banach space Y. The s ta tement that f is strongly differentiable at p E D means there is a continuous linear map T : X ---* Y such that if r > 0 there is a 5 > 0 such that if each of x and y is in D and is within 5 of p then I f (x) - f ( y ) - T ( z - Y)I <- r I x - Yl. In case D is admissible, T is unique and is denoted by i f (p) .

The s tatement that the ordered triple (X, D, V) is a local groupoid will mean that:

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i. D is a subset of the Banach space, X , containing 0, ii. Y : D x D ~ X satisfies Y(0, x) = V(x,O) = x for each x E D,

iii. V is s t rongly different;able at (0, 0) ; and iv. there is a b > 0 such tha t if each of x and y is in D and is within b of

0 then V ( x , y ) e D. In 1938, in the course of showing the mul t ip l icat ion V must be analytic ,

Birkhoff [3] gives the following theorem.

T h e o r e m 1. (Birkhoff, G. 1938) Suppose (X , D, V) is a local groupoid, D is open in X , V is associative, and f : [0,1] --* n satisfies f(O) = 0 and f is strongly different;able at O. Then there is a d > 0 such that if 0 < t < d

then l i rn ( f exists. Moreover, the canonical transformation T , defined

on some neighborhood of 0 by T (x ) = lira is strongly differentiable at 0

and T' (O) = I .

Thus, the Inverse Funct ion theorem guarantees tha t V may be t rans- formed into canonical parameters via the canonical t ransformation. More pre- cisely, there is a ne ighborhood D of 0 and a function T : D ~ X defined by

T(x ) = lira ( x ) n sat isfying T(0) = 0, T is s t rongly different;able at 0, and r t - - + ( ~ n -

T'(0) = I . Moreover, it follows tha t there is a ne ighborhood B of 0 such tha t if we define the function W : B x B ~ X by W ( x , y ) = T - I ( V ( T ( x ) , T ( y ) ) ) , then there is a c > 0 such tha t if x is within c of 0 and each of s, t , and s 4- t is in [ -1 , 1] then W ( s x , tx) = (s + t)x . Hence, the function T appl ied again to W yields the ident i ty t ransformation. I t is the function T , followed by the change in mul t ip l ica t ion W , which is the so called canonical t ransformat ion. Fur thermore , the canonical t ransformat ion appl ied to W also yields the ident i ty t ransformat ion. Such a mul t ip l ica t ion is said to be under canonical parameters .

If the hypothesis tha t D is open is weakened so tha t D is required only to be admissible, the existence of p roduc t integrals becomes unclear. This question, however, was answered affirmatively by the au thor in [1]. Moreover, it should be noted tha t the admiss ibi l i ty requirement may be weakened still fur ther so tha t D is s imply an a rb i t r a ry subset of X containing 0. (This may be of interest in opera to r theory, eg. [8], [9], [4].) However, the s t rong different;abil i ty of V at (0, 0), which may not yield a unique derivative if D is a rb i t r a ry must be s t rengthened to read: If r > 0 there is a 5 > 0 such tha t if each of w, x , y , and z is in D and is within 5 of 0, then IV(w, x) - Y ( y , z) - ((w - y) + (x - z))[ < e I(w - y, x - z) l . Tha t is, the derivative at (0, 0) is given by addit ion; which of course is a consequence in case D is admissible. We now consider the case where the associa t iv i ty hypothesis is weakened to power associativity. A proof of the following may be found in [5].

T h e o r e m 2. (Holmes, J.P. 1972) I f (X, D, V) is a local groupoid, D is open in X , and V is power associative, then there are positive numbers c and d such that i f x is in D and is within c of 0 then there is a unique one-parameter subsemigroup Tz from [0, 1] to the ball of radius d centered at O, B(0, d). That is, Tz : [0,1] --~ B(O,d) satisfies V ( T , ( s ) , T x ( t ) ) = T~(s + t) whenever each of s, t , and s + t is in [0,1], T~(O) = O, and T,(1) = x.

Al though this theorem gives us much informat ion on the s t ruc ture of power associat ive groupoids, the canonical pa ramete r s question, as discussed above and in recent papers by J. Kozma ([6], [7]), leads us once again to the

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investigation of the existence of product integrals. In particular, is there a d > 0 (~)" such that if x is within d of 0 then l i m exists? The author, in [21, sheds

some ligt~t on this in the finite dimensional case with the following theorem. (However, the existence of product integrals even in this case is left open.)

T h e o r e m 3. (1988) I f ( X , D , V ) is a local 9roupoid, D is locally compact and admissible, and V is power associative, then there is a e > 0 such that if x is within c of 0 and sx is in D for each s E [0,1] then some subsequence of {(g)~ '~ }n=a~176 converges. Moreover, if one constructs a one parameter subsemi- group, Tx , in the obvious manner, (given the convergent subsequence above) then T~ will be strongly diZerentiable at 0 and T'(0)(1) = x.

Recall that L~ : D --+ X is defined as left translation by y and for each positive integer n greater than 1 L~ is defined by x ~ V ( y , L ~ - a ( x ) ) . Let L~ : D ~ X denote the identity. The following theorem gives sufficient conditmns for the existence of product integrals in e~rbitrary dimension power associative groupoids.

T h e o r e m 4. Suppose (X, D , V ) is a local groupoid, D is a closed admissible subset of X , V is power associative, and f : [0, 1] --+ D satisfies f(O) = 0 and f is strongly differentiable at O. Suppose further that V satisfies condition (C1); that is, V satisfies:

There are positive d and M such that if each of x , z, and y is in D and each of x , z, and ny is within d of O, then

I L~(x) - L~(z) l < M i x - z h for each j E { 0 , 1 , . . . , n - 1}.

t n

Then there is a ~ > 0 such that if 0 < t < ~ then ~im ( f ( z ) ) _ e~sts

The proof to Theorem 4 follows from a short sequence of lemmas. proof to the following lemma may be found in [1].

A

L e m m a 5. I f e > 0 there is a 5 > 0 such that if each of x l , . . . , x n is

in D and ~ Ix,I < *, then fI x, is in D and I I~ x , - ~ x, I_<~ ~ Ix,I i = l i = l i = 1 i = 1 i = l

Here I] xi denotes V(xn , V ( x n - 1 , . . . , V(x2, Xl))'' "). Notice, this implies that i = 1

IN ~,1 < 2 ~ Ix, l provided O < ~ < 1. i = 1 i = 1

L e m m a 6 . There are B > 0 and O < s < l such that if O < t < s and each of m and n is a positive integer, then

i. Is(-~) l -< B'--., and L j f t m " f t ,n , - j - 1 i i I s~.~((( ( ~ ) ) ) " - ' ) - L ~ c . ~ ( v ( s ( ~ ) ' ( ( ( ~ - ) ) ) )1-<

t m 2MI (f(~--Z.)) - f ( ~ ) I , / o r each j E { 0 , 1 , . . . , n - 1}.

P r o o f . It follows from the strong differentiability of f at 0 that f is Lipschitz on some neighborhood of 0. Hence, let B > 0 and 0 < T < 1 such that if 0 < t < T then If(t)l < B t , which implies condition i. is satisfied. Using the strong differentiability of V at (0, 0) and the fact that V'(O, 0)(x, y) = x + y

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choose d>l > 0 so that if each of a, b, and c is in D and is within ~1 of 0, then IV(a,b) - V (c ,b ) - (a - c)l < ta - c[, which of course implies that IV(a, b ) - V ( c , b)l < 2 l a - c l . Next, choose 0 < ~ < 61 satisfying Lemma 5 for r =

3. C h o o s e O < s < m i n { a ~ , : - , T } a n d l e t O < t < s . I f j ~ { O , 1 , . . . , n - l } ,

it follows from the choice of s tha t r e ( n - j ) I f ( ~ - ~ ) ] _< r n ( n - j ) B ~ - - ~ < B s < ~. Hence, ] t m n - j 8 d ( ( f ( ~ - f f ) ) ) [ < 2 B s < min{~-, u Moreover, the same inequali ty

f t m n - - j �9 6 d holds for [ ( ( ( ~ - ~ ) ) ) [. Similarly, n [ f ( ~ ) l < Bs < m m { ~ , u Therefore, it follows from the choice of 6 and s tha t

] V ( f ( t ) , / rn~n--j--lX < t / + re\n--j--1

I \ n

d d < 2 ( ~ + ~ ) = e .

The conclusion now follows from condit ion (C1) and the choice of 61 �9

A proof to the following lemma may be found in [1].

___n L e m m a 7. I f e > O then there is a ~ > 0 such that i f ~ , [xil < t~ , then

i = 1

f x , ) - :(~,) <_ ~ i~,l i = 1 i = 1 i = l

We are now in a posi t ion to present a proof of Theorem 4.

P r o o f ( o f T h e o r e m 4) . I t suffices to show tha t if e > 0 there is a posit ive integer N such tha t if each of m and n is a posi t ive integer, n > N , then

f n t m n [ ( f i n ) ) -- (f(~"ffn)) I < e for sufficmntly smal l t . Therefore, choose B and s sat isfying Lemma 6 and let e > 0. Choose a posit ive integer N so tha t if each of m and n is a posit ive integer, n > N , and 0 < t < s then

i. Is(~)-mS(~) I <~' - ~-~ ~ < 4--M-'~, using Lemma 7, and

( ] ~ ) ) I -< ~mlS(m~) I < , ~ ~ < ,--~-, using r~emma ii. I m S ( ~ ) - - ' '~ " "' 5 and Lemma 6i.

I t then follows from the choice of s and N tha t if 0 < t < s, then

t n m n n

- - - - ( ( f ( ~ - - ~ n ) ) ) I

<_ 2 M n ( f t t _ l l) E

which completes a proof.

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Therefore, if the power associative multiplication V is defined on an

open neighborhood of 0 in X , then l i m exists for each x sufficiently

close to 0. Tha t is, the canonical transformation exists. In order to guarantee this transformation is also strongly differentiable at 0, and hence guarantee the multiplication can be transformed into canonical parameters, condition (C1) may be replaced by the following condition:

(C2) If c > 0 there i s a 6 > 0 such that if each of x, z, and ny is in D and is within 6 of 0, then

I L J ( x ) - LJv(z ) - (x - z)] < clx - z[ for each j E { 0 , 1 , . . . , n - 1}.

It should be noted that (C2) clearly implies (C1) by setting M = 1 + c, and thus the canonical transformation exists under condition (C2). Hence, we

X n define T : D ~ X by T ( x ) = ,-,oolim ( n ) . In order to show T is strongly

differentiable at 0, suppose ~ > 0. Choose 61 > 0 satisfying condition (C2) for c - - ~/9. Choose 0 < 62 < 61 such that if each of a, b, and c is in D a n d i s within 52 of 0, then ]V(a ,b) - V ( c , b ) - ( a - c)] _< ~ ] a - c]. Choose 0 < 53 < 52 satisfying Lemma 5 for 5 = 1. Finally, choose 6 > 0 such that 6 < ~- and suppose each of x and y is in D and is within 6 of 0. Choose a positive integer

( ) - ( ~ ) I<~1 -v l N such that if n is a positive integer, n >_ N then IT x , n * x

and I T ( y ) - ( ~ ) " ] < } I x - y ] . It follows from the choice of 6a that if n is a

positive integer then I~t < (52, ]~[ < 62, I ( ~ ) n - ' l < 21xl < 26 < a_.a., for each

a ~ {0,1, ,~}, and hence [V(L(~)~-J-1)] < 2(l~I + I (~)n-J-xl)< &. Thus, ] ( ~ ) " - J - V(~ , ( ~ ) , - j - 1 ) _ (~ _ ~)1 < ~l~ - ~], which also implies

I ( ~ ) n - j - v ( ~ , ( ~ ) n - j - 1 ) l - < 2 1 ~ - ~1" Hence, since 62 < 6 1 ,

<~ (g) - v ,( <9 ~- ; I . Therefore,

IT(x) - T ( y ) - (x - y)] X

g g

_< 51- - vl + 5 t~ - vI

+ ~ iJ ((x)n-j) -LJ (v(Y' (n)n- j - l ) ) - ( ~ - ~ ) 1

rL-1 , �9 j x n -a �9

- <)"- '- ' )) I - 2g

2e nae x y < o 7 1 ~ - vl + T ~ - _-_l = ~1- - vl.

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Thus, we have the following.

T h e o r e m 8. Suppose ( X , D , V ) is a local groupoid, D is open in X , V is power associative and satisfies condition (C2). Then the canonical transforma-

tion T : D --~ X defined by T(x) = lim ( x ) n exists, T is strongly differentiable n "--* O0 - n -

at 0 and T'(O)(x) : x . Hence, the inverse function theorem may be applied to transform V into canonical parameters.

Although condition (C1), which is used to show the existence of product integrals, is somewhat nonstandard, it is a consequence of the following more familiar condition used by Marsden [8] and Trotter [9]:

(C3) there are positive d and N such that if each of x and z is in D and is within d of 0 and 0 < t < d then IV ( f ( t ) , x ) - V ( f ( t ) , z ) l < e N t l x - - z I .

In order to indicate why condition (C1) is implied by (C3), let d and N satisfy (C3). U s i n g L e m m a 5 for ~ = 1, choose dl < d so that if each of x, z, and ny is in D and is within dl of 0 then ILJ(x)l < d and ILJ(z)[ < d for each j E {O, 1 , . . . , n } . Hence, letting f : [0,1] --- D defined by f ( x ) = s(ny) , it follows that if j E {0, 1 , . . . , n} then

n t ( x ) -- L t ( z ) : Y ( Y , n t - l ( x ) ) - V ( y , L t - I ( z ) )

j--1 " , < eN . (x) - L -l(z)

which, by induction on j is less than eN ~ (e N( ~"~ ) Ix -- zl ) < eNIx -- z I Setting M ---- e N completes a proof.

Clearly, a question of particular interest here is: Under what hypothesis will condition (C1) be a consequence? For example, if V is C O) and power associative and D is open, does (C1) follow in the infinite dimensional case, or at least the finite dimensional case? The answer is yes in the dimension 1 case.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

References

Anderson, M., One parameter submonoids in locally complete monoids, Colloquium Mathemat icum L I X (1990) 253-262.

Anderson, M., One parameter submonoids in locally compact differen- tiable monoids, Proc. Amer. Math. Soc. 106 (1989), 261-268.

Birkhoff, Garrett , Analytical Groups, Trans. Amer. Math. Soc. 43 (1938), 61-101. Chernoff, P., Semigroup product formulas and addition of unbounded operators, Bull. Amer. Math. Soc. 76 (1970), 395-398.

Holmes, J. P., Differentiable power associative groupoids, Pacific Journal Math. 41 (1972), 391-394.

Kozma, J., Behavior of loops in a canonical coordinate system, Arch. Math. 55 (1990), 498-502.

Kozma, J., On the differentiability of Ioopmultiplication in canonical co- ordinate system, Publ. Math. Debrecen 37 (1990), 313-325.

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[9]

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Marsden , J . , On product formulas for nonlinear semigroups, J. Fune. Anal . 13 (1973), 51-72.

Tro t t e r , H. F. , On the product of 8emigroups of operators, Proe . Amer . Math . Soc. 10 (1959), 545-551.

University of Hawaii at Hilo Hilo, Hawaii 96720-4091 e-mail MITCH~UHCCVX.BITNET

Received December 2, 1991 and in final form June 1, 1992

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