A groupoid approach to C* - algebras

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

793 Jean Renault

A Groupoid Approach to C*-Algebras

Springer-Verlag Berlin Heidelberg New York 1980

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

AMS Subject Classifications (1980): 22 D 25, 46 L 05, 54 H 15, 54 H 20 ISBN 3-540-09977-8 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-0997?-8 Springer-Verlag NewYork Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin Heidelberg 1980 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210

CONTENTS Page Introduction

I Chapter I : LOCALLY COMPACTGROUPOIDS

5

I.

Definitions and Notation

2.

Locally Compact Groupoids and Haar Systems

16

3.

Quasi-lnvariant Measures

22

4.

Continuous Cocycles and Skew-Products

35

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

5

47

1.

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

48

2.

Induced Representations

74

3.

Amenable Groupoids

86

4.

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

97

5.

Automorphism Groups, KMS States and Crossed Products

Chapter I I I

: SOME EXAMPLES

109

121

1.

Approximately-Finite Groupoids

121

2.

The Groupoids 0

138

n

Appendix : The Dimension Group of the GICAR Algebra

148

References

151

Notation Index

155

Subject Index

157

INTRODUCTION

The interplay between ergodic theory and von Neumann algebra theory goes back to the examples of non-type I factors which Murray and von Neumann obtained by the group measure construction [54].

A natural and probably d e f i n i t i v e point of view which

joins both theories has recently been exposed by P. Hahn [45].

I t uses the notion of

measure groupoid, introduced by G. Mackey "to bring to l i g h t and e x p l o i t certain apparently f a r reaching analogies between group theory and ergodic theory" ([53], p.187).

In p a r t i c u l a r , the group measure algebra may be regarded as the von

Neumann algebra of the regular representation of some principal measure groupoid. Moreover, most of the properties of the algebra may be interpreted in terms of the groupoid. The same standpoint is adopted by J. Feldman and C.Moore [31], in the framework of ergodic equivalence r e l a t i o n s .

Besides, they characterize abstractly

the von Neumann algebras arising from t h e i r construction. I t is natural to expect that topological l o c a l l y compact groupoids play a simil a r role in the theory of C*-algebras. The notions of topological and of Lie groupoid were introduced by Ehresmann for applications to d i f f e r e n t i a l topology and geometry. More recent i n t e r e s t in topological groupoids has come from the theory of f o l i a t i o n s ([10] ,p.273). I t seems to be the d i f f e r e n t i a l geometry point of view, rather than Mackey's v i r t u a l group point of view which aroused J. Westman's i n t e r e s t in groupoids and led him to the construction of convolution algebras of groupoids, first

in the t r a n s i t i v e (and l o c a l l y t r i v i a l )

case [75] and then in the non-transitive

principal case [77]. However the relevance to the theory of induced representations is also apparent in [ 7 ~ . Convolution algebras of transformation groups had already been used for some time [16,37]. The main works about transformation group C*-algebras,

by Effros and Hahn [24]

and by Z e l l e r - M e i e r [ 8 0 ] ,

appeared at about the same time

as Westman's a r t i c l e . Although t h e i r main purpose is to construct i n t e r e s t i n g examples of C * - a l g e b r a s , Effros and Hahn also give some results on the structure of a transformation group C * - a l g e b r a . This goal is more apparent in Z e l l e r - M e i e r ' s work, which is more d i r e c t l y motivated by group representation theory. Most of the l a t t e r work about transformation group C*-algebras concerns i t s e l f with the structure theory of these algebras ( f o r example [39]).

The s t a r t i n g point of t h i s work is a theorem of S. S t r ~ t i l #

and D.Voiculescu

about a p p r o x i m a t e l y - f i n i t e (or AF) C*-algebras [ 6 ~ . Generalizing the method of L.Garding and A. Wightman [3 4

f o r studying f a c t o r representations of the canonical

anticommutation r e l a t i o n s of mathematical physics, they show that every AF C * - a l g e b r a can be diagonalized and use a diagonalization to study i t s structure and i t s representations. In our s e t t i n g , t h i s amounts to saying that every AF C * - a l g e b r a is the C * - a l g e b r a of a principal groupoid (3.1.15). The construction (2.1) of the C * - a l g e b r a of a groupoid is modelled a f t e r the construction of the C * - a l g e b r a of a transformation group given by Effros and Hahn. Since a l o c a l l y compact groupoid does not necessarily have a Haar system, (Westman uses the term of l e f t

i n v a r i a n t continuous system of measures), needed to define

the convolution product, and since such a Haar system need not be unique, (although some r e s u l t s about existence and uniqueness of Haar systems can be found in K. Seda's articles

[67,68], we consider l o c a l l y compact groupoids with a f i x e d Haar system.

The case of r - d i s c r e t e groupoids, which generalize discrete transformation groups, deserves special a t t e n t i o n , because i t includes a l l our examples. An r - d i s c r e t e groupoid has a Haar system i f and only i f i t s range map is a local homeomorphism, and, i f t h i s is the case, i t is a scalar m u l t i p l e of the counting measures system ( 1 . 2 . 8 . ) . In the general case, but under suitable hypotheses, we show that the strong Morita equivalence class of the C *-algebra does not depend on the choice of the Haar system (2.2.11). The theory of group C*-algebras suggests many generalizations. In p a r t i c u l a r , one expects a correspondence between u n i t a r y representations of the groupoid and non-

degenerate representations of i t s C * - a l g e b r a . This is established (2.1.23) under a rather technical condition which w i l l often be needed, namely the existence of s u f f i c i e n t l y many non-singular Borel G-sets ( d e f i n i t i o n 1.3.27). I t is also possible to induce a representation from a closed subgroupoid (2.2.9). We give a d e f i n i t i o n of amenability in section 3 of chapter 2. I t develops that the C * - a l g e b r a of an amenable groupoid concides with the reduced C * - a l g e b r a , obtained by considering only the representations induced from the u n i t space (2.3.2).

Moreover, using some of

R. Zimmer's ideas about amenable measure groupoids [ 8 2 , 8 ~ , i t is e a s i l y shown that t h i s C * - a l g e b r a is nuclear (2.3.5). From our point of view, the most i n t e r e s t i n g groupoids are principal groupoids. Their C*-algebras

appear as genuine generalizations of matrix algebras. We have

looked for a characterization of these algebras s i m i l a r to the condition given by Feldman and Moore f o r algebras over an ergodic equivalence r e l a t i o n . The notion of Cartan subalgebra we give (2.4.13) is rather r e s t r i c t i v e and not as congenial as the corresponding notion f o r von Neumann algebras. In p a r t i c u l a r , we show by an example (3.1.17) that a regular maximal s e l f - a d j o i n t abelian subalgebra which is the image of a unique conditional expectation need not be a Cartan subalgebra. The correspondence between closed two-sided ideals of the reduced C * - a l g e b r a of a p r i n c i p a l groupoid and the closed i n v a r i a n t subsets of i t s u n i t space is established in the r - d i s c r e t e case (2.4.6). A continuous homomorphism (also called a one-cocycle) from a l o c a l l y compact groupoid to a l o c a l l y compact abelian group defines a continuous homomorphism of the dual group into the automorphism group of the C * - a l g e b r a of the groupoid (2.5.1). Moreover many one-parameter automorphism groups of the AF C *-algebras considered in mathematical physics (e.g. gauge automorphism group, dynamical groups) arise in t h i s fashion (examples 3.1.6 and 3.1.10). The groupoid point of view is p a r t i c u l a r l y well suited to t h e i r study. For example, the Connes spectrum of such an automorphism group is the asymptotic range of the cocycle (2.5.8) and the crossed product C * - a l g e b r a is the C * - a l g e b r a of the skew-product (2.5.7). Besides, the KMS condition f o r states may be replaced by a condition much closer to the o r i g i n a l Gibbs Ansatz characterizing e q u i l i b r i u m states (2.5.4). We use groupoids to derive p a r t i c u l a r but important cases

of some theorems of D. 01esen and G.K. Pedersen [5~ about s i m p l i c i t y and p r i m i t i v i t y of crossed product C~ - a l g e b r a s as well as the main results of O . B r a t t e l i

[~.

Another

a p p l i c a t i o n of groupoid C ~-algebras is the study of the C * - a l g e b r a of the b i c y c l i c semi-group and of the Cuntz C ~-algebras (3.2). A number of fundamental problems have not been touched in t h i s work. As we have seen e a r l i e r , groupoids have been introduced for two reasons. One is the " v i r t u a l group" point of v i e w , w e have not even given a d e f i n i t i o n of s i m i l a r i t y f o r l o c a l l y compact groupoids with Haar system. The other is the a p p l i c a t i o n to d i f f e r e n t i a l geometry, in p a r t i c u l a r to the theory of f o l i a t i o n s ; we have not made any mention of the work of A. Connes in t h i s d i r e c t i o n . These topics must await f u r t h e r development in the future. The author wishes to express indebtness to Marc Rieffel for numerous and f r u i t f u l suggestions and to Paul Muhly f o r a careful reading of the manuscript. He would l i k e to thank P. Hahn, who taught him about groupoid algebras and J.Westman f o r some unpublished material he gave him.

CHAPTER 1 LOCALLY COMPACTGROUPOIDS

The f i r s t

chapter sets up the framework of t h i s study. To gain some motivation

f o r the d e f i n i t i o n s which are given there, the reader can look simultaneously at the examples of the t h i r d chapter. I t is also useful to keep in mind the example of transformation groups, which is recalled below, and which suggests most of the terminology. The f i r s t

section gives the algebraic setting of the theory. The two main

concepts are groupoids and inverse semi-groups. The d e f i n i t i o n of a l o c a l l y compact groupoid with Haar system is introduced in the second section. The t h i r d section deals with the notion of q u a s i - i n v a r i a n t measure, and a generalization of i t ,

the

KMS condition, The results thereof w i l l be of great use in the second chapter. Some elementary properties o f one-cocycles are studied in the fourth section. Given a onecocycle, one can build the skew-product groupoid, and a basic question is to determine i t s structure in terms of the cocycle. An essential tool is the asymptotic range of the cocycle, which is the topological analog of Krieger's asymptotic r a t i o set in ergodic theory (see [ 3 ~ ,

I , d e f i n i t i o n 8.2).

1. D e f i n i t i o n s and Notation We shall use the d e f i n i t i o n of a groupoid given by P. Hahn in [4 4

(definition

1.1). I t is e s s e n t i a l l y the same as the one used by J.Westman in [7~ and the one used by A. Ramsay in [61].

1.1.

Definition :

A groupoid is a set G endowed with a product map (x,y) ~ xy :

G2 ÷ G where G2 is a subset of G x G called the set of composable pairs, and an

i n v e r s e map

x

~

-1

(x-l) -I

(i)

(x,y)

(ii) (iii)

(iv) If (x,y)

x

: G ~ G such t h a t

~ G2 =>

(y,z)

,

(xy,z),(x,yz)

E G2 and i f

(x,y)

c G2, then x -1 ( x y ) = y

(x,x -I)

c G2 and i f

(z,x)

E G2, then ( z x ) x -1

space o f G, i t s

Units will

usually

i s t h e domain o f x and r ( x )

= xx -1 i s i t s

the range o f y i s t h e domain o f x.

elements are u n i t s

i n the sense t h a t

be d e n o t e d by u, v , w w h i l e a r b i t r a r y

=

{x ~ G : x -I

AB

= {z E G :

E

GO = d(G) = r(G) xd(x)

one-to-one,

it

x ~ A, y ~ B : z = x y }

i s s a i d t o be t r a n s i t i v e ~ GO, GU = r - 1 ( u ) '

For u, v ,

G(u) = Gu ' which i s a g r o u p , The r e l a t i o n

elements will

denotes the orbit

u ~ v iff

be d e n o t e d

Examples :

a.

Transformation groups Suppose t h a t

groupoid structure = (u,st),

The map ( u , e )

if

t h e map ( r , d )

the map ( r , d )

the i s o t r o p y

from G i n t o

GO x GO i s

is onto.

group a t u.

Gu # @ i s an e q u i v a l e n c e V

orbits

relation

and the o r b i t

s is denoted u-s. : (u,s)

and ( v , t )

and ( u , s ) -1 = ( u . s , s - 1 ) . ~ u identifies

example. Then G i s p r i n c i p a l

transitively.

if

iff

on t h e u n i t

We l e t

it

has a s i n g l e

orbit.

G be U x S and d e f i n e

are composable i f f Then r ( u , s )

S acts freely,

GO/G

The image o f t h e

v = u.s

= (u,e)

GO w i t h U. The t e r m i n o l o g y iff

soace GO .

o f u i s denoted [ u ] .

t h e group S a c t s on t h e space U on the r i g h t .

p o i n t u by t h e t r a n s f o r m a t i o n

(u.s,t)

subsets o f G :

.

space. A g r o u p o i d i s t r a n s i t i v e

1.2.

from t h i s

= x.

Gv = d - l ( v ) ' Guv = GUn Gv and

is called

equivalence classes are called

(u's,e).

= x and r ( x ) x

is the

G}

A g r o u p o i d G i s s a i d t o be p r i n c i p a l

(u,s)

r a n g e . The p a i r

z.

A- I

following

:

= z

I f A and B are subsets o f G, one may form t h e f o l l o w i n g

Its

are s a t i s f i e d

c G2 and ( x y ) z = x ( y z )

(x-l,x)

x ~ G, d ( x ) = x - l x

by x , y ,

relations

= x

i s composable i f f

unit

the following

the

,

and d ( u , s )

of orbits

and t r a n s i t i v e

iff

=

comes S acts

b.

The groupoid G2 The s e t G2 of composable elements may be given the f o l l o w i n g groupoid s t r u c t u r e :

( x , y ) and ( y ' , z )

are composable i f f

y~ = xy, ( x , y )

( x y , z ) = ( x , y z ) , and ( x , y ) -1 =

(xy,y-Z). Then r 2 ( x , y ) = ( x , r ( y ) ) x

~ (x,d(x))

identifies

One may n o t i c e t h a t i t

= ( x , d ( x ) ) and d 2 ( x , y ) = ( x y , d ( x y ) ) .

The map

the u n i t space o f G2 w i t h G. The groupoid G2 is p r i n c i p a l . comes from the a c t i o n o f G on i t s e l f .

I t is t r a n s i t i v e

iff

G i s a group. c.

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

f o l l o w i n g groupoid s t r u c t u r e

: ( u , v ) and ( v ' , w ) are composable i f f

(v,w) = ( u , w ) , and ( u , v ) - I = ( v , u ) .

Then, r ( u , v )

v' = v, ( u , v )

= (u,u) and d ( u , v ) = ( v , v ) . The u n i t

space of R is the diagonal and may be i d e n t i f i e d

w i t h U. R is a p r i n c i p a l groupoid.

Conversely, i f G is a p r i n c i p a l g r o u p o i d , ( r , d )

identifies

G w i t h the graph o f the

equivalence r e l a t i o n ~. d.

Group bundle A group bundle G is a groupoid such t h a t fo

bundle is the union o f i t s iff

they l i e

u n i t space o f G i f f

1.3. D e f i n i t i o n

u c GO.

¢2 : G2

it

Given any groupoid G, G' = {x ~ G

the i s o t r o p y group bundle of G. I t is reduced to the

: Let G and H be groupoids. A map ~ : G ÷ H, i s a homomorphism ¢ ( y ) ) ~ H2 and

¢0 : GO ÷ H0 denotes the r e s t r i c t i o n

: G ÷ H are s i m i l a r

if

¢(x) ¢(y) = ¢ ( x y ) . Then ¢(u) ~ H0

÷ H2 i s the ma9 ¢ 2 ( x , y ) = ( ¢ ( x ) , ¢ ( y ) )

phisms ¢,¢

: d(x) = r ( x ) }

G is p r i n c i p a l .

f o r any ( x , y ) ~ G2, ( ~ ( x ) , if

A group

i s o t r o p y groups G(u). Here, two elements may be composed

in the same f i b e r .

is a group bundle. We c a l l

any x ~ G, d(x) = r ( x ) .

(write ¢ ~ ~)if

of

; it

¢ to the u n i t spaces. i s a homomorphism. Two homomor-

t h e r e e x i s t s a f u n c t i o n e : GO ÷ H

such t h a t ( e ~ r ) ( x ) ¢(x) = ~(x) ( e o d ) ( x ) f o r any x ~ G. Groupoids G and H are c a l l e d similar

( w r i t e G ~ H) i f

¢ o ~ and

t h e r e e x i s t homomorphisms ¢ : G ÷ H and ~ : H ÷ G such t h a t

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

isomorphisms.

Before g i v i n g a r e s u l t of Ramsay [61]

(theorem 1.7, p. 260) which i l l u s t r a t e s

t h i s n o t i o n , we need a d e f i n i t i o n .

1.4, D e f i n i t i o n

: Let G be a g r o u p o i d , E a subset o f GO ; GI = {x ~ G : r ( x ) e E ~E

and d(x)E E} i s a subgroupoid o f G w i t h u n i t space E ; GIE i s c a l l e d the r e d u c t i o n o f G by E.

1 . 5 . P r o p o s i t i o n : Let G be a g r o u p o i d , E a subset o f GO which meets each o r b i t

in

GO ," then GIE ~ G.

1.6. D e f i n i t i o n

: Let G be a g r o u p o i d , A a group and c : G ÷ A a homomorphism, the

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

x and y are composable and b = a c ( x ) ,

(x-l,ac(x)

; r(x,a)

= (r(x),a),d(x,a)

(x,a)(y,ac(x))

= (d(x),ac(x)).

A basic example o f skew-product is the f o l l o w i n g . the space U i n t o i t s e l f

= ( x y , a ) , and ( x , a ) -1 = I t s u n i t space is GO x A.

Let s be a t r a n s f o r m a t i o n o f

and l e t f be a f u n c t i o n on U w i t h values in anabelian group A.

On the space U x A, d e f i n e the t r a n s f o r m a t i o n t by ( u , a ) t = ( u s , a + 1~(u)). Let us d e f i n e the groupoid G o f s as the groupoid associated w i t h the corresponding t r a n s f o r m a t i o n group ( U , Z )

and d e f i n e s i m i l a r l y

the groupoid o f t . We leave to the reader

to check t h a t the groupoid o f t is the skew-product o f the groupoid G o f s by the homomorphism c : G -~ A o b t a i n e d from f by the r u l e s c(u,n) =

n-1 Z O

c(u,O) =

O, and

f ( u t i ) f o r n > 1,

c ( u , - n ) = - c ( u , n ) f o r -n < -1. Another i m p o r t a n t way o f b u i l d i n g up

new groupoids from o l d ones is the semi-

d i r e c t product.

1.7. D e f i n i t i o n

: Let G be a g r o u p o i d , l e t A be a group and l e t ~ : A + Aut(G) be a

homomorphism. We w r i t e x-a = ~ ( a - l ) ] product G x

(x) f o r a ~ A and x ~ G. The s e m i - d i r e c t

A i s the groupoid G x A where ( x , a ) and ( z , b ) are composable i f f

z = y-a

w i t h x and y composable, ( x , a ) ( y . a , b ) = ( x y , a b ) , and ( x , a ) -1 = (x -1 • a, a - l ) . Then, r ( x , a )

= (r(x),e)

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

w i t h GO.

An example of s e m i - d i r e c t product i s the groupoid associated w i t h a t r a n s f o r m a t i o n group (U,A).

In t h i s case G = U is reduced to i t s u n i t space. When G is a group,

1.7 is the usual n o t i o n o f s e m i - d i r e c t product. There i s a n a t u r a l a c t i o n o f A on the skew-product G ( c ) , namely the homomorphism defined by the formula m ( a ) ( x , b ) = ( x , a b ) and t h e r e is a n a t u r a l homomorphism c o f the s e m i - d i r e c t product G x

1.8. P r o p o s i t i o n : (i) (ii)

G(c) x (G x

A i n t o A, d e f i n e d by the formula c ( x , a ) = a.

With above n o t a t i o n , A is s i m i l a r t o G and

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

Proof : One may apply 1.5.

For example, to prove ( i ) ,

E = GO x {e} o f the u n i t space GO x A o f G(c) x f e r e n c e , l e t us w r i t e down e x p l i c i t l y (i)

Define

~ from G(c) x

by ~ ( x ) = ( x , e , c ( x ) )

C~

(G x

A meets each o r b i t .

the s i m i l a r i t y

homomorphisms :

and d e f i n e 0 from GO x A t o G(c) x

Define ~ from (G x

For f u r t h e r r e -

A to G by ~ ( x , a , b ) = x, d e f i n e ~ from G to G(c) x

check t h a t ~ o ~ (x) = x and e [ r ( x , a , b ) ] ( x , a , b ) (ii)

c~

one observes t h a t the subset

A by

C~

e ( u , a ) = ( u , e , a -1) and

= ~o~p ( x , a , b ) e [ d ( x , a , b ) ] .

A ) ( c ) t o G by ~ ( x , a , b ) = x - b -1

d e f i n e ~ from G t o

A ) ( c ) by ¢(x) = ( x , e , e ) and d e f i n e e from GO x A to (G x A ) ( c ) by 0 ( u , a ) =

(u • a - l , a , e )

and check t h a t ~ o ~(x) = x and

0[r(x,a,b)](x,a,b)

= ~o¢(x,a,b)

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

Together w i t h the n o t i o n o f g r o u p o i d , the n o t i o n o f inverse semi-group plays an i m p o r t a n t r o l e in t h i s work. The d e f i n i t i o n p r o p e r t i e s , can be found in [ 1 1 ] ,

given below, as w e l l as some elementary

page 28, or [ 1 ] .

A

10

1.9. D e f i n i t i o n

:

An i n v e r s e semi-group is a s e t ~

r y o p e r a t i o n , noted m u l t i p l i c a t i v e l y , following

relations

and an i n v e r s e map s ÷ s -1 : ~ ÷ ~

are satisfied:ss-ls

= s

Then t h e i n v e r s e map i s an i n v o l u t i o n . s and r ( s )

= ss -1 i s i t s

i n t o an i n f

The r e l a t i o n

such t h a t the

and s - l s s -1 = s - I .

If

s ~

, d(s) = s-ls

i s t h e domain o f

range. The set o f idempotent elements i s denoted by g O

idempotent elements commute. The r e l a t i o n which makes i t

endowed w i t h an a s s o c i a t i v e b i n a -

e ~ f iff

e f = e i s an o r d e r r e l a t i o n

Two on gO

semi-lattice.

between groupoids and i n v e r s e semi-groups is given by i n t r o d u c i n g

the n o t i o n o f G-set o f a g r o u p o i d . 1.10. Definition

:

L e t G be a g r o u p o i d . A subset s o f G w i l l

the r e s t r i c t i o n s

o f r and d to i t

be c a l l e d a G-set i f

are o n e - t o - o n e . E q u i v a l e n t l y ,

s i s a G-set i f f

ss - I

and s - I s a r e c o n t a i n e d i n GO.

Let g s e g => s

be the s e t o f G-sets o f G.

-1

~g.

We note t h a t s , t

=>

s t e g and

These o p e r a t i o n s make g i n t o an i n v e r s e semi-group. Note t h a t the

n o t a t i o n s d ( s ) and r ( s )

agree w i t h the p r e v i o u s ones.

A G-set s d e f i n e s v a r i o u s maps as f o l l o w s (i)

~g

f o r x on G w i t h d ( x ) ~ r ( s ) ,

:

the element xs o f G is d e f i n e d by { x s } = { x } s

( t h i s makes sense) ; (ii) (iii)

f o r x in G w i t h r ( x ) f o r u in r ( s ) ,

notations will

~ d(s),

the element sx o f G is d e f i n e d by { s x } = s { x }

the element u - s in d(s)

be used s y s t e m a t i c a l l y .

is d e f i n e d by u . s = d ( u s ) .

The map u ~ u • s : r ( s ) ÷ d(s) w i l l

;

These

be c a l l e d

t h e G-map a s s o c i a t e d w i t h the G-set s. The r e a d e r should not have any t r o u b l e t o check that ×(st)

= (xs)t

u • (st)

;

(ts)x = t(sx)

(xs) -1 = s-Zx - I

;

= (u . s) • t

where, w i t h our c o n v e n t i o n , x ( s t ) t and s i m i l a r l y

;

(ts)x

is d e f i n e d by { x ( s t ) }

is d e f i n e d by { ( t s ) x }

= ts{x}.

= {x}st

f o r the G-sets s and

11

To help understanding what G-sets mean, l e t us look a t the case o f a t r a n s f o r mation group (U,S). Any element s o f the group S d e f i n e s the f o l l o w i n g G-set o f the associated groupoid G : s = { ( u , s )

: u e U}. I t s domain and i t s range are U. The

associated G-map i s the t r a n s f o r m a t i o n u ~ u • s and t h e r e i s no a m b i g u i t y in the n o t a t i o n s . The map from S to the set o f G-sets above d e f i n e d is an inverse semi-group homomorphism. I t i s one-to-one but u s u a l l y not onto. Note t h a t in the case o f a group, t h a t i s , when U is reduced to one p o i n t , the G-sets are e x a c t l y the elements o f S. J. Westman has developed in [ 7 ~

a cohomology theory f o r groupoids which extends

the usual group cohomology t h e o r y ; i t

i s reproduced here.

Suppose t h a t C i s some c a t e g o r y . A map p from a set A onto a s e t A0 such t h a t each f i b e r called

p-l(u)

is an o b j e c t o f C

C-bundle.

will

be c a l l e d a C - b u n d l e map and A w i l l

be

For example, a group bundle in the sense o f 1 . 2 . d i s a C - b u n d l e

where C is the category of groups and any such C - b u n d l e i s a group bundle. Let A be a C - b u n d l e w i t h bundle map p : A ÷ AO. Write Au = p - l ( u ) . #u,v : Av ÷ Au : u , v , E AO} are composable i f f

Iso(A) = {isomorphisms

has a n a t u r a l s t r u c t u r e of groupoid : #u,v and ~ v ' , w

v' = v - then t h e i r product i s ~u,v °#v,w' and 4 -1 ' U~V

morphism inverse of ~u,v" The b i j e c t i o n

idu, u ~ u identifies

i s the i s o -

the u n i t space o f Iso(A)

and AO. Iso(A) i s c a l l e d the isomorphism groupoid o f the C - b u n d l e A. 1.11. D e f i n i t i o n

:

Let G be a g r o u p o i d . A G-bundle

(A,L) i s a C - b u n d l e A t o g e t h e r

w i t h a homomorphism L : G ÷ I s o ( A ) such t h a t L0 : GO ÷ A0 is a b i j e c t i o n . often identify

GO and AO). When C i s

(We w i l l

the category of a b e l i a n groups, one speaks o f a

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

d e f i n e Gn f o r any n ~ N. The sets G0, GI = G and G2 have a l r e a d y been d e f i -

ned. For n ~ 2, Gn i s the set of n-uples (x 0 . . . . . Xn_l) c Gx...xG such t h a t f o r i = 1..... n-l,

x i i s composable w i t h i t s l e f t

from Gn to A which s a t i s f i e s (i) (ii)

neighbor. A n-cochain i s a f u n c t i o n f

the c o n d i t i o n s

p o f ( x 0 . . . . . Xn_l) = r(XO) and i f n > 0 and f o r some i = O , . . . , n - 1 ,

x i c GO, then f ( x 0 . . . . . x i . . . . . Xn_l)

12 E A0 . The set Cn(G,A) of n-cochains is an abelian group under pointwise a d d i t i o n . The an> n+l sequence 0 ÷ cO(G,A) ÷ CI(G,A) . . . . . Cn(G,A) - - C (G,A) . . . . . where ~Of(x) = n L(x) f~d(x) - f o r ( x ) and a n ( f ( x 0 . . . . . Xn) = L ( x o ) f ( x I . . . . . Xn) + Z (-1) i i=l f ( x 0 . . . . . x i _ i x i . . . . . Xn_l) + (-1) n+l f ( x 0 . . . . . Xn_l) f o r n > O, is a cochain complex. 1.12. D e f i n i t i o n :

The group of n-cocycles of t h i s complex w i l l

the group o f n-coboundaries w i l l Zn(G,A)/Bn(G,A) w i l l

be denoted by Zn(G,A),

be denoted by Bn(G,A)and the n-th cohomology group

be denoted by Hn(G,A).

A section f o r a G-bundle (A,L) is a f u n c t i o n f from A0 to A such t h a t pof(u) = u, where p is the bundle map. A section f is said to be i n v a r i a n t i f L(x) fod(x)= f o r ( x ) f o r every x ~ G. The set of sections w i l l

be denoted by F(A) and the set of

i n v a r i a n t sections by I~G(A). I f (A,L) is a G-module bundle, cO(G,A) = F(A) and HO(G,A) = rG(A ). A one-cocycle c ~ ZI(G,A) is a one-cochain f from G to A which s a t i s f i e s f ( x y ) = L(x)f(y) + f(x).

In p a r t i c u l a r ,

i f A is a constant bundle, t h a t i s , each f i b e r Au is

equal to a f i x e d a b e l i a n group B, and i f G acts t r i v i a l l y

on A, t h a t i s , L(x) is the

i d e n t i t y map of B f o r every x, a one--cocycle f c ZI(G,A) is a homomorphism of G i n t o B. In the case of a constant bundle A as above w i t h t r i v i a l

a c t i o n , we w r i t e ZI(G,B)

instead of ZI(G,A). We may also consider one-cocycles with values in a not necessarily a b e l i a n group. In t h i s case, (A,L) is a G-bundle where A is a group bundle. We define ZI(G,A) = {f : G ÷ A : f(xy) = f(x)[L(x)f(y)]}, such t h a t f ( x ) = [b o r ( x ) ] - l [ L ( x ) b o d ( x ) ] } f ~, g i f f

BI(G,A) = { f

: G ÷ A : there e x i s t s b : GO ÷ B

and the equivalence r e l a t i o n on ZI(G,A)

there e x i s t s b : GO ÷ B such t h a t f ( x ) = [ b o r ( x ) ] - 1

:

(x) [ L ( x ) b o d ( x ) ] .

As f o r groups, two-cocycles are r e l a t e d to groupoid extensions : 1.13. D e f i n i t i o n :

Let (A,L) be a G-module bundle, noted m u l t i p l i c a t i v e l y .

An

extension of A by G is an exact sequence of groupoids A0 ÷ A-~i>E~J>G ÷ GO (we also w r i t e ( E , i , j ) ) compatible w i t h the a c t i o n of G on A, in the sense t h a t there e x i s t s a section k f o r j such t h a t

13 (i)

k(u) = u

(ii)

k(x) i ( a )

(AO, E0 and GO are i d e n t i f i e d ) k(x) -1 = i ( L ( x ) a

Two extensions ( E , i , j )

and ( E ' , i

phism # : E ÷ E' such t h a t i '

f o r any ( a , x ) e A x G w i t h p(a) = d ( x ) . ,j')

are e q u i v a l e n t i f

= #oi and j = j ' o # .

t h e r e e x i s t s an isomor-

The set of e q u i v a l e n t classes o f

extensions w i t h the Baer sum is an a b e l i a n group denoted Ext(A,G). 1.14. P r o p o s i t i o n : H2(G,A) = Ext(A,G). Sketch o f the p r o o f :

Given

~ ~ Z2(G,A), l e t E

= {(a,x)

c A x G : p(a) = r ( x ) } .

o

I t s groupoid s t r u c t u r e is given by ( a , x ) and ( b , y ) are composable i f f (a,x)(b,y)

x and y are ; then

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

and ( a , x ) - I = ( ( m ( x - l ) a - 1 ) ~ ( x - I , x ) - l , x - I ) . Define i ( a ) = ( a , p ( a ) ) and j ( a , x ) section.

It

is r e a d i l y v e r i f i e d

= x and note t h a t k(x) = ( r ( x ) , x ) that (E

,i,j)

is a c o v a r i a n t

is an extension and t h a t i t s class

depends only on the class o f o. Conversely, i f

(E,i,j)

is an extension of A by G and k is a c o v a r i a n t s e c t i o n ,

then ~ defined by i ( ~ ( x , y ) )

= k ( x ) k ( y ) k ( x y ) -1 is a 2-cocycle in Z2(G,A). I t s class is

not a f f e c t e d by another choice o f s e c t i o n or an e q u i v a l e n t e x t e n s i o n . F i n a l l y @: (a,x) ~ i(a)k(x)

: E ~ E sets up an equivalence of E and E.

The t r i v i a l

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

Let us f i n a l l y coefficients

note t h a t two s i m i l a r groupoids have same cohomology groups w i t h

in a t r i v i a l

constant module bundle. E x p l i c i t l y ,

H ÷ G be two h a l f - s i m i l a r i t i e s

l e t # : G ÷ H and ~ :

; ~o~ ~ id G and ~o ~ ~ id H. The maps f

Cn(G,A) ~ Cn(H,A) and g ~ go#n :

÷ fo~n :

Cn(H,A) -~ Cn(G,A) give isomorphisms of the cohomo-

logy groups. A cohomology t h e o r y f o r inverse semi-groups may be given along the same l i n e s . Suppose t h a t C

is some c a t e g o r y . Let A0 be a s e t . The set 2AO o f a l l

when ordered by i n c l u s i o n , V c U. A C - s h e a f

subsets of AO,

is a category : there is an arrow V ÷ U p r e c i s e l y when

A based on A0 is a c o n t r a v a r i a n t f u n c t o r U ÷ A U on 2AO t o C

(the

14 morphism "~U ~J{V corresponding to V c U should thought of as the r e s t r i c t i o n A partial of A0

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

together with isomorphisms # :~{V'

the r e s t r i c t i o n

÷~(V')'

f o r any V ' c

morphisms, t h a t i s , such t h a t f o r V " c

mutes

Two p a r t i a l

morphism).

iV'

* ~(V').

~Vl '

' ~t (~)( Vii )

V, compatible with

V', the f o l l o w i n g diagram com-

isomorphisms # and #' may be composed : we have # : V -~ U and #' : V' ÷ U' ;

we l e t V" be #,-1 (U'n V) and U" be #(U'n V) ; #" = #o#' is the b i j e c t i o n V" ÷ U" obtained by composing # and #' ; and f o r V c V" we define #" :J{V_ ÷ ~ # " ( V ) posing~v

~ ''> j { # , ( V~) <

# o# ,(V). The inverse of a p a r t i a l

by com-

isomorphism is defined in

the obvious fashion. These operations make jso(Y~) = { p a r t i a l

isomorphisms of~4} i n t o

an inverse semi-group, that we c a l l the isomorphism inverse semi-group of the C - s h e a f s { . 1.15. D e f i n i t i o n

:

Let ~ be an inverse semi-group. A g - s h e a f (~,£) is a C-sheaf

together w i t h a homomorphism £: g ÷ Jso(~) such t h a t [0 : gO ÷ 2Ao is an i n j e c t i o n . We l e t gn be g x . . . x g

n times f o r n > 1 and gO be as before. Given a g - s h e a f

(~{, £) of abelian groups, one can form the f o l l o w i n g cochain complex. A n-cochain is a f u n c t i o n f from g n t o ~ { which s a t i s f i e s (i) (ii)

f(So,S I . . . . . Sn_1) e ~{

r(SoS 1 .--Sn_ 1) ;

f is compatible with the r e s t r i c t i o n

and V = r ( t 0 t I . . . f(to'tl

the conditions

t n _ l ) where t i = eis

maps, t h a t i s , i f U = r(s 0 s I . . . S n _ l )

f o r some idempotent element e i then

. . . . . tn-1) ~ ~ V is the r e s t r i c t i o n

(ill)

of f ( s ,s . . . . . . s l ) c ~ . , to V ; and Ao ± n-~ u '0 f o r n • O, f ( s 0 . . . . . s i . . . . . Sn_l) ~ 2 whenever s i is an idempotent element.

The set c n ( g , J { )

of n-cochains is an abelian group under pointwise a d d i t i o n . The

sequence 0 + cO(g,~)

~ cZ(g,~)

.....

f od(s) -

+ cn(g,.~)

where

6Of(s) = £ ( s )

and

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

~n> cn+l ( ~ , ~ ) .......

fo r(s)

n

+ i!i

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

+ (-1) n+l is a cochain complex.

f(s 0 ....

Sn_l),

÷

15 1.16, D e f i n i t i o n : The group of n-cocycles and the group of n-coboundaries of this complex w i l l be denoted respectively by z n ( ~ , ~ ) logy group z n ( g , ~ ) / B n ( g , ~ )

and by B n ( ~ , ~ ) .

The n-th cohomo-

w i l l be denoted H n ( g , ~ ) .

Before giving the next d e f i n i t i o n , l e t us remark that ~ = u ~ U, where U runs overdO = 2AO, has a structure of inverse semi-group, where for a C~U and

b c ~ V,

a + b is the element o f ~ UnV obtained by adding up the r e s t r i c t i o n s of a and b to UnV. 1.17. D e f i n i t i o n :

Let ( ~ , £) be a g-sheaf of abelian groups, noted m u l t i p l i c a t i -

vely. An extension o f ~ by ~ is an exact sequence of inverse semi-groups y~O ÷j~ i> 8

j> g ÷

~0

(we also write ( ~ , i , j ) )

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

(i)

k(e) = e

(ii)

for e ~ gO

(~cO 80 and gO are i d e n t i f i e d )

k(s) i ( a ) k(s) - I = i ( £ ( s ) a )

(iii)

for (a,s) ~ J~x~.

k(es) = ek(s) and k(se) = k(s)e

Two extensions ( 8 , i , j )

and ( 8 ' , i ' , j ' )

for

e c ~0

s ~.

are equivalent i f there exists an isomorphism

: 8 + 8' such that i ' = #oi and j = j'o@. The set of equivalence classes of extensions with the Baer sum is an abelian group denoted Ext ( ~ , ~ ) a n d j u s t as before, Ext ( ~ , ~ )

is isomorphic to H2(j~, ~ ) .

1.18. F i n a l l y , we note the relationship between the cohomology of a groupoid G and the cohomology of the inverse semi-group of i t s G - s e t s , ~ .

Let (A,L) be a G-module

bundle. One forms the following ~ - s h e a f of abelian groups based on AO, ( ~ , £ ) .

For

UcAO,J~U = {sections of A defined on U} with i t s additive structure ; for VcU, the morphism ~ defined by : V cd(s)

÷~

is the usual r e s t r i c t i o n map. The homomorphism £ : ~ ÷ Jso(d~) is

£(s) is the b i j e c t i o n d ( s ) ÷

and U = Vs- I C

r(s),

£(s) : ~ ' V

r(s) which sends u into u . s - I and for ÷~

is given by £(s) h(u) = L(us) h(u . s)

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

Namely

f(So,S 1 . . . . . Sn_l) is the section of A defined on r(s 0 s I . . . Sn_l) by f(So,S I . . . . . Sn_l) (u) = f(USo,(U • So)SI . . . . . (u - SoS1 . . . Sn_2)Sn_l). I t is compat i b l e with the r e s t r i c t i o n maps. The map f

~ f commutes with the coboundary opera-

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

16

(resp Bn ( ~ , ~ ) ) .

Conversely, given g E c n ( ~ , ~ ) ,

f(Xo,X I . . . . . Xn_l) =

g ( { x } , {x I } . . . . . {x n 1 } ) 0

~ Bn(G,A) ; H n ( ~ , ~ )

(r(XO)) where { X o } , { x 1} . . . . . ~x n 1) are

-

considered as G-sets. Then g = f . Bn(~,~)

we may d e f i n e f ~ Cn(G,A) by

~

In conclusion c n ( ~ , ~ )

~ Hn(G,A). We w i l l

-

~ Cn(G,A) ; zn(~,~) ~ zn(m,A) ;

use a t o p o l o g i c a l v e r s i o n o f t h i s

r e s u l t in 2.14.

2. L o c a l l y Compact Groupoids and Haar Systems.

The d e f i n i t i o n found in [ 7 9 ] ,

o f a t o p o l o g i c a l groupoid and i t s

[26] page 23 and [68] page 26.

2.1. D e f i n i t i o n

:

A t o p o l o g i c a l groupoid c o n s i s t s of a groupoid G and a t o p o l o g y

compatible w i t h the groupoid s t r u c t u r e (i)

x ~ x

(ii)

immediate consequences can be

-i

:

: G ~ G is continuous

( x , y ) ~> xy : G2 -- G is continuous where G2 has the induced t o p o l o g y from

G x G. Consequences :

x ~ x

-1

,

is a homeomorphism ; r and d are continuous ; i f

G is

Hausdorff, GO is closed in G ; i f GO is Hausdorff, G2 is closed in G x G. GO is both a subspace of G and a q u o t i e n t o f G (by the map r) tient

; the induced t o p o l o g y and the quo-

topology coincide. We w i l l

only consider t o p o l o g i c a l groupoids whose t o p o l o g y is Hausdorff and,

w i t h the e x c e p t i o n of s e c t i o n 4, l o c a l l y theory o f i n t e g r a t i o n on l o c a l l y I f X is a l o c a l l y

compact. We w i l l

u s u a l l y use B o u r b a k i ' s

compact spaces [ 5 , 6 , 7 ] .

compact space, Cc(X ) denotes the l o c a l l y

convex space of

complex-valued continuous f u n c t i o n s w i t h compact s u p p o r t , endowed w i t h the i n d u c t i v e limit

topology.

2.2.

Definition

: Let G be a l o c a l l y

compact groupoid. A l e f t

c o n s i s t s of measures {~u, u e GO} on G such t h a t (i)

the support supp ~u of the measure ~u is Gu,

Haar system f o r G

17 (ii) (iii)

( c o n t i n u i t y ) f o r any f ~ Cc(G),u (left

~

~(f)(u)

= ffd~ u is continuous, and

i n v a r i a n c e ) f o r any x ~ G and any f ~ Cc(G),

ff(xy)

d~d(X)(y) =

f f(Y)dAr(X)(y). This is Westman's d e f i n i t i o n

([77] p.2) of a l e f t

o f measures. I t d i f f e r s from Seda's d e f i n i t i o n

i n v a r i a n t continuous system

([68] p . 2 7 ) . i n two respects : no

measure on the u n i t space is given and c o n t i n u i t y is required ; t h i s l a s t assumption is a r a t h e r severe r e s t r i c t i o n

on the topology of G.

In Section 4 of [68] and

theorem 2 of [67], Seda gives c o n d i t i o n s under which c o n t i n u i t y holds a u t o m a t i c a l l y ; i t seems p r e f e r a b l e here to assume i t

as p a r t of the d e f i n i t i o n .

The f o l l o w i n g r e s u l t s are easy consequences of the d e f i n i t i o n 2.3.

Proposition :

2.4.

P r o p o s i t i o n : Let G be a l o c a l l y compact groupoid with a l e f t

(cf.[77]

1.3 , ! . 4 ) .

~: Cc(C ) -~ Cc(GO) is a continuous s u j e c t i o n . Haar system.

Then r : G -~ GO is an open map, and the associated equivalence r e l a t i o n on the u n i t space is open. 2.5.

Examples : !

(a)

A l o c a l l y compact t r a n s f o r m a t i o n group G = U x S has a d i s t i n g u i s h e d l e f t

Haar system :

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

f o r S. (b)

I f G is a l o c a l l y compact groupoid, then G2 with the topology induced from

G x G is also a l o c a l l y compact groupoid. I f { u} is a l e f t

Haar system f o r G, then

{(~2)x} is a l e f t Haar system f o r G2 where ff

d(~2) x = f f ( x , z )

d~d(x)

(z) f o r f ~ Cc(G2).

For example, i f G is a group, G2 = G x G. As a groupoid, i t with the t r a n s f o r m a t i o n group (G,G) where G acts on i t s e l f Haar system is 6x x ~, where ~ (c)

is a l e f t

is the groupoid associated by t r a n s l a t i o n .

Its left

Haar measure f o r G, as in example a.

Let G be a l o c a l l y compact p r i n c i p a l groupoid. The map d : Gu ~ [u] is a

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

for a left

Haar system on G is :

18

a system o f measures { m [ u ] ' u c GO} where (i) (ii)

m[u] is a measure on

[u] of support [u]

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

set of GO x GO). These d e f i n i t i o n s are e q u i v a l e n t : i f and s a t i s f i e s

(i')

system, vlhere

and ( i i ' )

{Xu } is g i v e n , ~[u] = d ~ U

; conversely i f

depends only on [u]

{ ~ [ u ] } is given, {~u} is a l e f t

Haar

ffd~ u = f f ( u , v ) d m [ u ] ( v ) .

(d) Let G be a l o c a l l y compact group bundle, t h a t i s , a l o c a l l y compact groupoid which is a group bundle in the sense of 1.2.d.

Then a l e f t

is e s s e n t i a l l y unique in the sense t h a t two l e f t

Haar system, i f

it exists,

Haar systems {~u} and {v u} d i f f e r

by

a continuous p o s i t i v e f u n c t i o n h on GO : xu = h(u) u. The i s o t r o p y group bundle G' = {x e G : d(x) = r ( x ) } of a l o c a l l y compact groupoid G is closed, hence l o c a l l y compact. In the case where G is a t r a n s f o r m a t i o n group, the existence of a l e f t system on G' is the assumption made in [ 3 ~

(see beginning of the f i r s t

886) to determine the t o p o l o g i c a l s t r u c t u r e of the space of a l l

Haar

section page

i r r e d u c i b l e induced

representations of G. (e)

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

compact Hausdorff space when equipped with F e l l ' s x c K} c S x G with the topology induced from (K,x) and (L,y) are composable i f f

topology [32]. G = {(K,x)

: K E S,

S x G and the groupoid s t r u c t u r e :

K = L, ( K , x ) ( K , y ) = (K,xy),

(K,x) -1 = (K,x -1) is a

l o c a l l y compact group bundle, t h a t we may c a l l the subgroups bundle of G. I t is shown in ~2] t h a t a l e f t Haar system (~K) e x i s t s . K c S, ~K is a l e f t 2.6.

I t is e s s e n t i a l l y unique by d. For each

Haar measure f o r K.

D e f i n i t i o n : A l o c a l l y compact groupoid is r - d i s c r e t e i f

i t s u n i t space is an

open subset. 2.7.

Lemma : (i) (ii) (iii)

Let G be an r - d i s c r e t e groupoid.

For any u e GO, Gu

and Gu are d i s c r e t e spaces.

I f a Haar system e x i s t s , i t

is e s s e n t i a l l y the counting measures system.

I f a Haar system e x i s t s , r and d are l o c a l homeomorphisms.

19

Proof :

(i)

An x in Gv d e f i n e s a homeomorphism y ~ xy : Gv ÷ Gu - since #v} is ooen U

in Gv, { x } (ii)

~

•

"

'

is open in Gu. Let {~u} be a l e f t

Haar system. Since Gu i s d i s c r e t e and ~u has support Gu,

every p o i n t in Gu has p o s i t i v e ~U-measure. Let g = ~,(×GO) , where XG0 is the c h a r a c t e ristic

f u n c t i o n of GO. I t

is continuous and p o s i t i v e .

may assume t h a t ~U({u}) X

~

Replacing ~u by g(u)-1~ u, we

= 1 f o r any u. Then by i n v a r i a n c e , ~ V ( { x } ) =

1 f o r any

Gv . u (iii)

We assume, as we may, t h a t xu is the counting measure on Gu. Let x be a

p o i n t of G. A compact neighborhood V o f x meets Gu in f i n i t e l y

many p o i n t s x i

i = 1 . . . . . n. I f x i ~ x, t h e r e e x i s t s a compact neighborhood V' of x contained in V, which does not c o n t a i n x . . T h e r e f o r e , we may assume t h a t GunV = { x } . Then ~ r ( x ) ( v ) = 1. l

By c o n t i n u i t y o f the Haar system, we may assume t h a t ~U(v) = 1 f o r any u ~ r ( V ) . This shows t h a t r 2.8.

:

V ÷ GO is i n j e c t i v e ,

Proposition :

For a l o c a l l y

hence a homeomorphisms onto r ( V ) .

compact groupoid G, the f o l l o w i n g p r o p e r t i e s are

equivalent : (i) (ii) (iii) (iv)

G is r - d i s c r e t e

and admits a l e f t

Haar system,

r : G + GO i s a l o c a l homeomorphism, the product map G2 ÷ G i s a l o c a l homeomorphism, and G has a base o f open G-sets.

Proof : (i)

~>

(ii)

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

(ii)

~>

(iii)

If

( x , y ) E G2, we may choose a compact neighborhood U o f x and

a compact neighborhood V o f y such t h a t r l v and d i v are homeomorphisms onto t h e i r I

images ; U x V n G2 i s then a compact neighborhood of ( x , y ) on which the product map is i n j e c t i v e . x'y'

= x"y" = > r ( x ' ) and d ( y ' )

(iii)

~

(iv)

= r(x")

~ > x' = x"

= d ( y " ) = > y'

= y".

I f x E G and U is a neighborhood o f x, we may f i n d open sets V

and W such t h a t x e V c U, x -1

W c U-1 and the r e s t r i c t i o n

o f the product map to

20 V x W is i n j e c t i v e . SoV n W-1 is the d e s i r e d open G-set. ( i v ) =-=>( i i )

Clear.

( i v ) -~->(i)

The groupoid G i s r - d i s c r e t e

G-set s such t h a t u e r ( s ) = s s - l c

: f o r any u c GO, t h e r e i s an open

GO and by ( i i i )

ss -1 is open in G.

Let ~u be the counting measure on Gu and f be in Cc (G). Using a p a r t i t i o n identity,

one can w r i t e f as a f i n i t e

Therefore i t

o f the

sum o f f u n c t i o n s supported on open G-sets s,

is enough to consider a f u n c t i o n f whose support is contained in an open

G-set s. Then ~ ( f ) ( u )

= ~u(f) =

f(us)

: ~(f)

is continuous. Q.E.D.

2.9, C o r o l l a r y : system i f f

A locally

compact groupoid G i s r - d i s c r e t e and admits a l e f t

G2 i s r 2 - d i s c r e t e and admits a l e f t

2.10. D e f i n i t i o n

:

Haar

Haar system.

Let G be an r - d i s c r e t e groupoid. I t s ample semi-group ~ i s

the

semi-group o f i t s compact open G-sets. This t e r m i n o l o g y , introduced by W. Krieger in [5 4 , end of the s e c t i o n . The case of i n t e r e s t

will

be j u s t i f i e d

at the

is when G admits a cover o f compact open

G-sets. Then G has a base of open G-sets, w i t h sub-base {Us : U open subset o f GO and s c ~ } ,

t h e r e f o r e G admits a l e f t

Haar system. We do not know i f

there e x i s t r -

d i s c r e t e groupoids which have a Haar system but do not have a cover o f compact open G-sets. I f G has a cover of compact open G-sets, i t

is c o m p l e t e l y described by (GO, ~ )

in the sense t h a t i t s groupoid s t r u c t u r e as w e l l as i t s t o p o l o g y may be recovered from GO, ~ and the map r .

I f x ~ s, w i t h s e ~ ,

x c s, y c t w i t h s, t E ~ and d(x) = r ( y ) ,

x -1 is d e f i n e d by s -1 { r ( x ) }

xy is defined by { x y } = { r ( x ) } s t .

j u s t seen t h a t {Us : U open subset of GO, s E ~ } Let us d e s c r i b e next the r - d i s c r e t e

= x-I.

If

We have

is a sub-base f o r the t o p o l o g y o f G.

p r i n c i p a l groupoids which admit a cover o f

compact open G-sets. 2.11. D e f i n i t i o n

:

Let U be a l o c a l l y

compact space and s a p a r t i a l

homeomorphism

o f U, d e f i n e d on a compact open subset r ( s ) onto a compact open subset d ( s ) . say t h a t s is r e l a t i v e l y

free if

i t s set o f f i x e d p o i n t s {uc r ( s )

(compact and) open. Let us say t h a t an inverse semi-group ~ o f

Let us

: u • s = u} is

partial

homeomorphisms

21 defined on compact open subsets of U acts r e l a t i v e l y

f r e e l y i f each s E ~ is r e l a -

tively free. 2.12. D e f i n i t i o n :

Let U be a l o c a l l y compact space and ~ a n inverse semi-group o f

p a r t i a l homeomorphisms defined on compact open subsets o f U.

Let us say t h a t ~ is

ample i f (i)

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

(ii)

f o r any f i n i t e

and d ( s i ) n d ( s j )

family (si)

= ~ for i # j,

i=1 . . . . . n in

~ such t h a t r ( s i ) n r ( s j )

=

there e x i s t s s in ~ d e n o t e d by ~s i such t h a t u • s =

u • si for u ~ r(si). 2.13. P r o p o s i t i o n :

Let U be a l o c a l l y compact space and ~ an inverse semi-group

o f p a r t i a l homeomorphisms

defined on compact open subsets of U. Let G be the

p r i n c i p a l groupoid associated w i t h the equivalence r e l a t i o n u ~ v

iff

there e x i s t s s ~ ~ : u = v • s

Then the f o l l o w i n g p r o p e r t i e s are e q u i v a l e n t . (i)

G has a s t r u c t u r e o f r - d i s c r e t e groupoid with a cover o f compact open

G-sets such t h a t U becomes i t s u n i t space and i t s ample semi-group is the ample i n verse semi-group generated by ~ . (ii)

~

acts r e l a t i v e l y

f r e e l y on U.

Proof : (i) ~> (ii)

Let s and t be two compact open G-sets of G. Then s n t is a com-

pact open G-set of G. Thus, i f s ~ {u ~ r ( s ) (ii)

:>

: u • s = u}

(i)

= s n r(s)

s compact open in GO = V.

For s ~ ~ , l e t s = {(u,us)

: u c r(s)}.

gy which has as sub-base {Vs : V open in U and s ~ 3 } .

We d e f i n e on G the t o p o l o -

I t makes G i n t o a r - d i s c r e t e

groupoid a d m i t t i n g a cover o f compact-open s e t s , n a m e l y ~ . T h e induced topology on GO=u is i d e n t i c a l to the o r i g i n a l one. F i n a l l y , l e t s be a compact open G-set. I t may be covered by f i n i t e l y i = 1..... n in~

many open G-sets in ~ .

Hence there e x i s t s a f i n i t e

family (si)

and a f i n i t e

f a m i l y (Ui) i = 1 . . . . . n o f compact open sets o f U such n t h a t Ui n Uj = ~, Ui • s i n Uj • sj = ~ f o r i # j and s = U Uis i . i=l Q.E.D.

$2 2.14.

In the t o p o l o g i c a l s e t t i n g , we make the f o l l o w i n g adjustments to the cohomolo-

gy theory given in the f i r s t (a)

section (see [79] p.24).

In 1.11, we r e q u i r e t h a t A be a l o c a l l y compact group bundle and we re-

q u i r e t h a t f o r any continuous section u ~ au of p : A-~ AO, the f u n c t i o n x ~ L(X)ad(x) should be continuous. (b)

We give to Gn the topology induced from the product topology on Gx...xG

n-times and consider continuous cochains only. I t w i l l

be i m p l i c i t

t h a t Zn(G,A),

Bn(G,A) and Hn(G,A) r e f e r to the continuous cohomology. I f G is an r - d i s c r e t e groupoid which admits a cover o f compact open G-sets, the r e s u l t s o f 1.18 are s t i l l of G. Given g ~ c n ( ~ , ~ )

v a l i d when ~ is i n t e r p r e t e d as the ample semi-group

( n o t a t i o n s of 1.18), we define f e Cn(G,A) by

f(Xo,X 1 . . . . . Xn_l) = g(So,S 1 . . . . . S n _ l ) ( r ( x o ) ) where So,S 1 . . . . . Sn_I ~ a n d xI

~ s I . . . . . Xn_ 1 ~ Sn_1. By assumption, there e x i s t

x0 ~ sO,

sO, s I . . . . . Sn_1 w i t h these

p r o p e r t i e s . Moreover, the c o n d i t i o n t h a t g be compatible with the r e s t r i c t i o n shows t h a t f is well defined. F i n a l l y f is continuous since i t s r e s t r i c t i o n sO x SlX...XSn_ 1 is continuous. Thus Hn(G,A) ~

maps to

Hn(~,~).

3. Q u a s i - l n v a r i a n t Measures

Let G be a l o c a l l y compact groupoid with l e f t be the image of 3.1.

Haar system {~u}. Let ~u = ( ~ u ) - I

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

Definition :

Let u be a measure on GO. The measure on G induced by ~ is

= f~Ud~(u). The measure on G2 induced by ~ is 2 by the inverse map is - 1

= f~ux~U d~(u). The image o f

= f~ud~(u)"

These measures are well defined since the system {~u} o f measures on G the system

{~u×~U}

o f measures on G2 are ~-adequate (Bourbaki [6] 3.1) ; v

the measure on G2 induced by

-1

with respect to the Haar system 2.5.b.

2

and is also

23 3.2.

Definition :

A measure ~ on GO is said to be m u a s i - i n v a r i a n t i f

measure ~ is e q u i v a l e n t to i t s inverse - I .

i t s induced

A measure belonging to the class of

is also q u a s i - i n v a r i a n t ; we say t h a t the class is i n v a r i a n t .

I f G is second countable and ~ is a q u a s i - i n v a r i a n t measure on GO, then (G,C), where C is the class o f

~, is a measure groupoid in the sense of P.Hahn [44] p.15

and (v,u) is a Haar measure f o r (G,C) ( d e f i n i t i o n

3.11 p. 39). Most of the r e s u l t s

and techniques o f t h i s section can be found in [44] and in [61]. The cohomology theory f o r measure groupoids is developed in [76] ; the d i s c r e t e p r i n c i p a l case is studied thoroughly in [31]. The r e l e v a n t f a c t here is t h a t to each q u a s i - i n v a r i a n t measure is associated a 1-cocycle with values in fR~, whose class depends on the measure class o n l y . 3.3.Proposition :

Let ~ be a q u a s i - i n v a r i a n t measure on GO and D a l o c a l l y ~ - i n t e -

grable p o s i t i v e f u n c t i o n such t h a t v = Dv- I , (i)

for 2

a.e. ( x , y ) ~D(xy) = D(x)D(y) and

f o r ~ a.e. x~ D(x - I ) = D(x) - I (ii)

then

i f u' = g~

;

where g is a l o c a l l y ~ - i n t e g r a b l e p o s i t i v e f u n c t i o n , D' =

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

(i)

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

; this D2 ( x , y ) = D(xy)D(x) - I are versions of the Radon-Nikodym d e r i v a t i v e ~ d~2 (dv2) -1 gives the f i r s t a s s e r t i o n . (ii)

Straightforward. Q.E.D.

This p r o p o s i t i o n shows t h a t the Radon-Nikodym d e r i v a t i v e of ~ with respect to -i

(defined

V

a . e . ) is a one-cocycle With values i n l R : in the sense o f [76] §3

and t h a t i t s class depends on the class of ~ 3.4.

Definition :

only.

Let ~ be a q u a s i - i n v a r i a n t measure on GO ," (a version o f ) the

Radon-Nikodym d e r i v a t i v e D = dv is c a l l e d the modular f u n c t i o n (or the Radon-1 d~

24 Nikodym d e r i v a t i v e ) o f

u.

I f G is a group, the p o i n t mass a t e i s , up to a s c a l a r m u l t i p l e , q u a s i - i n v a r i a n t measure on GO =

{e}.

the only

I t s modular f u n c t i o n in the sense o f 3.4 equals

a . e . the modular f u n c t i o n o f the group. It will

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

symmetric measure

in the class of u , where symmetric means equal to i t s i n v e r s e (the i n v e r s e o f a measure on G is i t s

image under the inverse map). We choose ~O = D-1/2 ~ and c a l l

the symmetric measure induced by 3.5.

Definition

it

u.

: Let ~ be a q u a s i - i n v a r i a n t measure on GO. A measurable set A in

GO is almost i n v a r i a n t

( w i t h respect to

u) i f

f o r v a.e. x , r ( x )

c A iff

d(x)

~ A.

The measure ~ is c a l l e d e r g o d i c i f every almost i n v a r i a n t measurable set is null or conull. Let X and Y be l o c a l l y I f X is

q-compact, it

compact spaces and p a continuous map from X onto Y.

is p o s s i b l e to d e f i n e the image p . C o f a measure class C on X :

one chooses a p r o b a b i l i t y measure ~ in the class o f C and defines p . C o f p . ~, where p . ~ ( E ) = ~ ( p - l ( E ) )

as the class

; p , C depends only on the class o f C. As i t

is

e a s i e r to deal w i t h measures r a t h e r than w i t h measure c l a s s e s , one i~roduces the n o t i o n o f pseudo-image o f a measure (see [ 6 ] )

: a pseudo-image o f an a r b i t r a r y

X is a measure in the image p . C o f the class C o f 3.6.

Proposition :

induced measure v .

Let p b e a measure on GO and

measure u on

p. [u]

be a pseudo-image by d o f the

Then

(i)

[~] i s a q u a s i - i n v a r i a n t

; and

(ii)

~ is quasi-invariant iff

~ ~ [~].

Proof :

(i) [v](f)

Let Iv] = /~v d [ p ] ( v ) and f be a non-negative measurable f u n c t i o n . = O iff

for

[~] a.e. v, ~ v ( f ) = 0 ;

iff

f o r v a.e.

x and

iff

f o r p a.e.

u,

~d(x) a . e .

~u a.e.

x and

y, f(y)

= 0 ;

~d(x) a.e.

u, f ( y )

= 0

Then

25

iff

for ~ a.e.

u,

~u a . e . x and x-1~ u a . e .

iff

f o r u a.e.

u,

~u a.e. x and ~u a.e.

z, f ( x - l z )

= 0 ;

iff

f o r u a.e.

u,

xu a.e. x and ~u a . e .

z, f ( z - l x )

= O,

by F u b i n i ' s

(ii) 3.7.

If

Definition

a saturation of 3.8.

for u a.e.

u,

iff

f o r v a.e.

x and

iff

[v]-l(f)

pseudo-image o f

(ii) (iii)

~d(x) a.e. y , f ( y - 1 )

Y, f ( y - 1 )

= 0 ;

= 0 ;

~is

a pseudo-image by d o f

Let ~ be a measure on GO. Then a measure [~]

v-l~ as above is c a l l e d

u.

Proposition

(i)

~u a . e . x and ~d(x) a . e .

= O.

quasi-invariant, :

= 0 ;

theorem ;

iff

uis

y, f(y)

:

Let mu be the s a t u r a t i o n o f the p o i n t mass at u, t h a t i s , a

~u. Then

the class o f

mu depends o n l y on the o r b i t

[u]

;

mu is ergodic ; and every q u a s i - i n v a r i a n t

measure c a r r i e d by [u] is e q u i v a l e n t to ~u'

Proof : (i)

Let N be a subset o f [u] and v be in [u]

x c Gu d -1 (N) is ~ U - n e g l i g i b l e i f f v' (ii) (e.g.

The e r g o d i c i t y

it

. Since ~u = x .~v f o r

is ~ V - n e g l i g i b l e .

of a transitive

quasi-invariant

measure is w e l l known

[ 6 1 ] , theorem 4 . 6 , p. 278). Suppose t h a t A is almost i n v a r i a n t

and has p o s i t i v e

measure and l e t v be S~v dmu(V ). Then 0 = v [ d - l ( G ~ a ) ~V[d-l(GOA)] (iii)

n r-Z(A)]

= SA ~V[d-1(GOA)]dmu(V ). Hence, f o r some v in A,

= 0 and by ( i ) ~u(GO~A) = O. (cf.

[61]

measure such t h a t

, lemma 4 . 5 , p. 277). Let u be a q u a s i - i n v a r i a n t

~ ([u])

= I and l e t ~ be i t s

probability

induced measure on G. Then mu is a

pseudo-image o f ,~ by d : v[d-l(A)]

: 0

iff

for ~ a.e.

v,

~V[d-l(A)]

= 0 ;

iff

for u a.e.

v,

~U[d'l(A)]

= 0

iff

au(A) = 0 •

because o f ( i )

;

26

But so is u because of quasi-invariance : ~[d-Z(A)] = 0

iff

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

iff

f o r ~ a.e.

iff

~(A) = O.

v, ~v [ d - l ( A ) l = 0 ;

Q.E.D. I f G is the groupoid of a t r a n s i t i v e

transformation group (U,S), the class of

~u is the unique i n v a r i a n t measure class on U. This case is well known (e.g.

[74],

theorem 8.19, p.25). 3.9,

Definition :

A t r a n s i t i v e measure is a q u a s i - i n v a r i a n t measure carried by an

o r b i t . Up to equivalence, there e x i s t s one and only one t r a n s i t i v e measure on the orbit

[u]; i t w i l l be denoted ~[u]" A q u a s i - i n v a r i a n t ergodic measure which is not

t r a n s i t i v e is called properly ergodic. A q u a s i - o r b i t is an equivalence class of quasii n v a r i a n t ergodic measures. 3.10. Proposition :

Suppose that G is second countable. The modular function D of

the t r a n s i t i v e measure ~[u] can be chosen such that DIG(v ) = modular function of G(v) f o r ~ [ u ] a . e . v . Proof :

This is in theorem 4.4 p. 48 of [44]. An a l t e r n a t e proof is to use a s i m i l a -

r i t y between the e s s e n t i a l l y t r a n s i t i v e groupoid (G,~ru]) and the group G(u) (cf.

I69], theorem 6.19).

3.11. A well known theorem of J. Glimm [36] states t h a t , f o r a second contable l o c a l l y compact transformation group G, the f o l l o w i n g properties are equivalent : (i) (ii) (iii)

every o r b i t is l o c a l l y closed ; the o r b i t space GO/G with the quotient topology is TO ; every q u a s i - o r b i t is t r a n s i t i v e .

We do not know i f t h i s can be generalized to a r b i t r a r y second countable l o c a l l y compact groupoids with Haar system. The implications ( i ) ~ > ( i i ) obtained as in [36].

:>(iii)

may be

27 3.12. D e f i n i t i o n :

An i n v a r i a n t measure is a q u a s i - i n v a r i a n t measure whose modular

function is equal to 1. 3.13. D e f i n i t i o n :

([55] p. 448). Suppose G p r i n c i p a l . A q u a s i - o r b i t is called

( i ) type I i f i t is t r a n s i t i v e , (ii) (iii)

type 111 i f type I I

i t is properly ergodic and contains a f i n i t e

i n v a r i a n t measure,

i f i t is properly ergodic and contains an i n f i n i t e

invariant

measure, and ( i v ) type I I I 3.14, D e f i n i t i o n :

i f i t is properly ergodic and contains no i n v a r i a n t measure. A principal groupoid is of type I i f i t has type I q u a s i - o r b i t s

only. The notion of i n v a r i a n t measure can be extended as f o l l o w s . Before g i v i n g the d e f i n i t i o n , recall that ZI(G, IR) is the group of continuous homomorphisms of G into

IR.

Let c be in ZI(G, IR), then we denote by Min(c) the set of u's in GO such that C(Gu)iS in[O,~) and by Max(c) the set Min ( - c ) . 3.15, D e f i n i t i o n :

Let c ~ ZI(G,

IR) and

B ~ [-~, +~]. We say that a measure u on

GO s a t i s f i e s the (c,B) KMS condition i f ( i ) when ~ is f i n i t e ,

u is q u a s i - i n v a r i a n t and i t s modular function D is equal

to e -Bc . (ii)

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

p r o b a b i l i t y measure is also called a ground state f o r c. The point mass at u is called a physical ground state i f ~in(c) n [u] = The terminology w i l l be j u s t i f i e d

{u}.

in the section 4 of the second chapter.

However the condition D = e-~c is closer to the classical Gibbs Ansatz f o r e q u i l i b r i u m states than to the a n a l y t i c form of the KMS conditions (cf. example 3.1.6). 3.16. Proposition : (i)

Note f i r s t

that c ' l ( o )

is a l o c a l l y compact groupoid. I f G is r - d i s c r e t e

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

Suppose that G is r - d i s c r e t e and that B is f i n i t e .

measure f o r G is an i n v a r i a n t measure f o r c-1(0).

Then, a (c,B) KMS

28

(iii) (iv)

The subset Min(c) is closed in GO. The subset Min(c) is i n v a r i a n t under c - 1 ( 0 ) , t h a t i s , i f x ~ c-1(0)

and d(x) c H i n ( c ) , then r ( x ) c Min(c). (v) Proof : (iii)

The reduction of G to Min(c) is equal to the reduction of c-1(0) to Min(c). Assertions ( i ) and ( i i )

are c l e a r .

I f u ~ M i n ( c ) , there e x i s t s x E G such t h a t d(x) = u and c(x) < O. Let V

be an open neighborhood o f c such t h a t c(y) < 0 f o r y ~ V. Then d(v) is an open neighborhood of u and d(V) n Min(c) = @ . (iv)

Let x E c-~1(0) w i t h d(x) ~ N i n ( c ) . For any y c Gr(x),

yx E Gd(x) and

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

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

= O.

Q.E.D. 3.17. Proposition :

(cf.

[65], theorem 7.5, page 26)

A limit

p o i n t ( w i t h respect to

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

Suppose t h a t ~B tends to u as B tends to ~

f u n c t i o n of ~ sure of

is e -Bc. Let vB

u. Then v~

and suppose t h a t the modular

be the induced measure and l e t v be the induced mea-

tends to v and vB-1 tends to - 1

f o r every non-negative f in Cc(G ), ffcd~ -1 = lim = lim

as ~ tends to

~. Therefore,

ffcd~B-I f f c e ~c dv~

= lim (fc > 0 fce~C d ~ + Jc < 0 fceBc d~B)" Since ce ~c tends to 0 u n i f o r m l y on c < O, the second i n t e g r a l

tends to O. Hence

f f c d ~I is non-negative f o r every non-negative f c Cc(G ). Thus, c is non-negative on the support of v -1, which is d - l ( s u p p u ) . That i s , suppu

is

contained in Min(c). Q.E.D. The l a s t part of t h i s section is devoted to the study of the r e l a t i o n s h i p between the notion of q u a s i - i n v a r i a n c e given in 3.2. and the usual notion of q u a s i - i n v a r i a n c e

2g

under an inverse semi-group of t r a n s f o r m a t i o n s . Let us f i r s t

look a t the case o f a t r a n s f o r m a t i o n group (U,S). Let G = U x S

be the associated groupoid. The measure on G induced by the measure ~ on U is v = x ~, where ~ i s a l e f t S acts in two d i f f e r e n t (i)

Haar measure o f S. With respect to the groupoid G, the group ways :

The h o r i z o n t a l a c t i o n i s the a c t i o n o f S on U. One says t h a t ~ i s q u a s i -

invariant if (ii)

it

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

The v e r t i c a l

a c t i o n i s the a c t i o n o f S on i t s e l f ,

o r r a t h e r on each f i b e r

{u) x S. One notes t h a t ~ i s q u a s i - i n v a r i a n t under t h i s a c t i o n . right,

dx -s - I d~

I f we l e t S a c t on the

is equal to ~ ( s ) , where a is the modular f u n c t i o n o f S.

Before studying the general case, l e t us e s t a b l i s h some conventions : Let (X,u) and ( Y , v ) be two measure spaces and s : X ÷ Y a bimeasurable b i j e c t i o n

from X onto Y.

The image o f x by s i s w r i t t e n x - s and the image o f ~ by s i s w r i t t e n ~ - s. Thus, ~ f ( y ) d ( ~ • s ) ( y ) = ~ f ( x • s ) d u ( x ) f o r f E Cc(Y ). I f u " s is a b s o l u t e l y continuous w i t h respect to v, d~ .s denotes the Radon-Nikodym d e r i v a t i v e o f u" s w i t h respect to v. dv One says t h a t s is n o n - s i n g u l a r i f i t induces an isomorphism o f the measure a l g e b r a s .

3.18.

Definition

:

Let G be a l o c a l l y

compact groupoid w i t h Haar system {~u}. Let

be a measure on GO, not n e c e s s a r i l y q u a s i - i n v a r i a n t ,

and v be i t s

induced measure.

Let s be a G-set measurable w i t h respect to the completion o f v. (i) to v ) i f

We say t h a t ~ is ~ u a s i - i n v a r i a n t under s (or s is n o n - s i n g u l a r w i t h respect the map from ( d - l [ d ( s ) ] , V l d _ l [ d ( s ) ] )

i s non s i n g u l a r . -1 d~s The Radon-Nikodym d e r i v a t i v e T ( w h e r e

to ( d ' l [ r ( s ) ] ,

V l d _ Z [ r ( s ) ] ) defined by

the r u l e x ~ xs - I

restriction)

will

be denoted by

we w r i t e

~ instead o f the a p p r o p r i a t e

~( • ,s) and c a l l e d the v e r t i c a l

Radon-Nikodym

d e r i v a t i v e o f s ( w i t h respect to v ) . (ii) (r(s),

We say t h a t ~ i s q u a s i - i n v a r i a n t under s i f ~ir(s))

derivative

the map from ( d ( s ) , u l d ( s ) )

to

defined by the r u l e u ~ u • s -1 is n o n - s i n g u l a r . The Radon-Nikod~1 -I d~. s w i l l be denoted by A ( - , s ) c a l l e d the h o r i z o n t a l Radon-Nikodym

d e r i v a t i v e o f s ( w i t h r e s p e c t to ~ ) .

30 Remark :

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

space. Therefore, i f s is a measurable G-set, r ( s ) is measurable and the map from r ( s ) to s sending u to us is measurable. 3.19.Proposition

:

With the notations of the previous d e f i n i t i o n ,

assume that u is

q u a s i - i n v a r i a n t . Then the v e r t i c a l Radon-Nikodym d e r i v a t i v e of a non-singular measurable G-set s with respect to v depends on d(x) only. More p r e c i s e l y , there e x i s t s a function u ~ 6(u,s) defined on r ( s ) ,

p o s i t i v e and measurable and which we s t i l l

call

the v e r t i c a l Radon-Nikodym d e r i v a t i v e of s, such that ~ ( d ( x ) , s ) = d~s'l dv (x) f o r v a.e. x in d - Z [ r ( s ) ] . Proof : Let

a(x) = d(vs-1) (x) be the v e r t i c a l Radon-Nikodym d e r i v a t i v e of s with -

d~

respect to ~. Since vs " I = f d ( s )

(~Us-1) dr(u) and vs - I = f d ( s ) ~ ( ~ u )

d~(u) are two

r-decompositions of vs -1, there e x i s t s a ~-conull set U in GO such that f o r every u in U, ~u

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

~(x) = d(~Us-1)(x). d~u The commutativity of l e f t and r i g h t m u l t i p l i c a t i o n allows us to w r i t e , f o r any x in GU and any p o s i t i v e measurable f , f f ( y ) ~ ( x y ) d~d(X)(y) = f f ( x - l y )

{ ( y ) d~r(X)(y)

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

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

Hence, f o r any x in GU and ~d(x) a.e. y, ~(xy) = ~(y). Therefore, i f ¢ is a p o s i t i v e measurable function such that

~(u) :

f~(X)

f@d~u= 1 f o r u in U, the function 6 defined in r ( s ) by

~(X) dZu(X )

has the required property. Indeed, since U is r-l(u) is

~-conull, d-l(u) is v-l-conull and

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

v-l-conull,

hence ~u-COnull for ~ a.e.u. Thus, for u a.e. u and any positive measurable f, ~f(y) 6 od(y) d~U(y) = =

~f(y) ~(x) #Ix) d~d(y)(X) d~U(Y), #f(y) ~(xy)

= l(~f(Y)

~(xy)

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

=

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

=

#f(y)

#(y) dZU(y) ; therefore

31

If(y)

6od(y) dv(y) = ~f(y) ~(y) dv(y), = I f ( y s - I ) dv(y). Q.E.D.

3.20.Proposition

: Let u be a q u a s i - i n v a r i a n t

measure, v i t s induced measure and s a

measurable G-set. Then the following properties are equivalent

:

(i)

u is q u a s i - i n v a r i a n t

under s.

(ii)

u is q u a s i - i n v a r i a n t

under s. Moreover, i f these conditions are s a t i s f i e d ,

the v e r t i c a l ~(.,s)

and the horizontal

and A ( - , s ) ,

Radon-Nikodym derivatives

of s with respect to

are related by the equation ~(u,s) = D(us)

in r ( s ) , where D is the modular function of

u,

A(u,s) f o r ~ a.e. u

~.

Proof : Suppose that ( i ) holds. Given a non-negative measurable function h defined on r ( s ) ,

there exists a non-negative measurable function

such that h(u) = I f ( x )

d~u(X ) for u e r ( s ) ( c f

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

d~

h defined on d - l [ r ( s ) ]

2.3). Then,

-1 (x) d~(u) u.s

i f ( x s " I ) d~ u (x) d~(u)(by r i g h t invariance of {~u })

=

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

=

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

: If(x)

dv(x)

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

= [h(u) D-I(us) a(u,s) d~(u). Hence u is q u a s i - i n v a r i a n t

under s and d(~s-1) du (u) = D-l(us)~(u,s)

f o r ~ a.e. u in

r(s). Conversely, suppose that ( i i )

holds. Then, f o r any non-negative measurable func-

tion f defined on d - l [ r ( s ) ] , I f ( x s -1) dv(x) = I f ( x s -1) D(x) d~u(X ) du(u) = If(x) = If(x)

D(xs) d~

_l(X) d~(u)Iby r i g h t invariance of {~u}~ us D(xs) d~u(X ) A(u,s)dp(u)

=

If(x)

D(d(x)s) A(d(x),s)

D(x) dv-Z(x)

=

If(x)

D(d(x)s)

d~(x).

A(d(x),s)

32 This shows t h a t v is q u a s i - i n v a r i a n t under s and t h a t -1 dvs (x) = D ( d ( x ) s ) A ( d ( x ) , s ) f o r v a . e , x in d - l [ r ( s ) ] . Q.E.D. 3.21,Case o f a t r a n s f o r m a t i o n group. Let us look back to the case o f a t r a n s f o r m a t i o n group (U,S). With above n o t a t i o n , G = U x S and ~u = 5u x ~ where ~ is a l e f t and any G-set s = {u,u • s) : u E U} v

Haar measure of S. For any measure u on U

where s is an element o f S, the induced measure

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

o f S. I t is known ( e . g . [ 6 1 ] , sense o f 3.2 derivative

iff

it

theorem 4 . 3 , page 276) t h a t ~ is q u a s i - i n v a r i a n t in the

is q u a s i - i n v a r i a n t under the group S. The h o r i z o n t a l Radon-Nikodym

A(u,s) is the usual Radon-Nikodym cocycle o f the a c t i o n .

invariant,

it

D(u,s)

:

If ~ is quasi-

f o l l o w s from 3.20 t h a t i t s modular f u n c t i o n i s

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

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

groupoid.

Since the counting measure ~u is i n v a r i a n t under any G-set s, the v e r t i c a l Radon-Nikod3an 6 ( u , s ) is i d e n t i c a l l y

equal to 1, independently of any measure u on GO.

Suppose t h a t G admits a cover o f compact open G-sets and l e t

~ b e i t s ample semi-group

(definition

iff

under ~ .

2 . 1 0 ) . Then a measure u on GO is q u a s i - i n v a r i a n t Indeed i f ~ is q u a s i - i n v a r i a n t ,

quasi-invariant.

since any compact set can be covered by f i n i t e l y

Let X be a l o c a l l y

is q u a s i - i n v a r i a n t

by 3.20 any compact open G-set leaves

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

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

it

under 3 ,

it

is q u a s i - i n v a r i a n t

many compact open G-sets.

groupoid.

compact space. As in 1 . 2 . c , the graph X x X o f the t r a n s i t i v e

equivalence r e l a t i o n on X ( t h a t i s , any two elements o f X are e q u i v a l e n t ) has a s t r u c t u r e o f g r o u p o i d . With the product t o p o l o g y , i t

is a l o c a l l y

compact groupoid.

As in 2 . 5 . c , any measure on X w i t h support equal to X d e f i n e s a Haar system on X x X. The t r a n s i t i v e

measure

X induces the product measure m x m. A measurable G-set s is n o n - s i n g u l a r w i t h respect t o m x m i f f

it

(X,m). The h o r i z o n t a l and v e r t i c a l

is the graph o f a n o n - s i n g u l a r t r a n s f o r m a t i o n o f Radon-Nikodym d e r i v a t i v e s o f s w i t h respect to

33 are equal : 1

A(x,s) =

~(X,S) = d~s-~ (x) f o r ~ a.e. x in r(s) d~ The measure m is i n v a r i a n t because i t s modular f u n c t i o n is i d e n t i c a l l y equal to 1. We have defined in 3.18 the n o t i o n of a n o n - s i n g u l a r measurable G-set w i t h respect to the induced measure ~ o f a measure u on GO. I t w i l l

be useful to have a

d e f i n i t i o n depending o n l y on the groupoid G an the Haar system {~u}. 3 . 2 4 , D e f i n i t i o n : Let G be a l o c a l l y compact groupoid. (i)

A G-set s w i l l

restriction

be c a l l e d a Borel G-set [resp. a continuous G-set I

i f the

of each of the maps r and d to s is a Borel isomorphism onto a Borel

subset of GO [resp. a homeomorphism onto an open subset of GO]. (ii)

Suppose t h a t G has a Haar system {xu}. A n o n - s i n g u l a r Borel G-set

[resp.

non-singular continuous G-set] is a Borel G-set [resp. a continuous G-set] such t h a t there e x i s t s a Borel [resp. continuous]

p o s i t i v e f u n c t i o n on r ( s ) bounded above

and below on compact sets, denoted ~ ( - , s ) and c a l l e d the v e r t i c a l

Radon-Nikodym

d e r i v a t i v e of s, such t h a t ~ ( d ( x ) , s ) = d~Us--~l (x) f o r every u E GO and ~u a.e. x c d - 1 [ r ( s ) ] . d~ u Thus, a non-singular Borel G-set s is non s i n g u l a r w i t h respect to the induced v

measure

of every measure ~ on GO and dvs -1 (x) f o r v a.e. x c d - l [ r ( s ) ] . ~(d(x),s) = d~

3.25. Examples :_ In the case of a t r a n s f o r m a t i o n group (U,S), the G-set s = {(u,u • s) : u ~ V } where V is an open subset of U and s E S, is a n o n - s i n g u l a r continuous G-set. I t s v e r t i c a l

Radon-Nikodym d e r i v a t i v e is ~(u,s) = 6(s) f o r u c V,

where ~(s) the modular f u n c t i o n of S evaluated at s. In the case of a r - d i s c r e t e groupoid, any open G-set s is a n o n - s i n g u l a r continuous G-set. We have already observed t h a t i t s v e r t i c a l

Radon-Nikodym d e r i v a t i v e 5(u,s) is equal to 1, f o r u c r ( s ) .

3.26. The set o f n o n - s i n g u l a r Borel G-sets [ r e s p . n o n - s i n g u l a r continuous G-sets]

is

an inverse semi-group under the operations ( s , t ) ÷ st and s ÷ s -1. We c a l l i t the Borel ample semi-group o f G and denote i t and w r i t e

~c ] .

~ b [resp. the continuous ample semi-group of G

Let us note the f o l l o w i n g formulas : f o r s , t e ~ b J

34

~(u,st)

= 6(u,s)

~(u-s,t)

6 ( u , s -1) = l ~ ( u - s - l , s ) I

3.27. D e f i n i t i o n

:

"1

for u c r(st) for u c d(s)

Let G be a l o c a l l y

say t h a t G has s u f f i c i e n t l y

compact groupoid w i t h Haar system. We w i l l

many n o n - s i n g u l a r Borel G-sets i f

f o r e v e r y measure u

on GO w i t h induced measure v on G, e v e r y Borel set in G o f p o s i t i v e c o n t a i n s a n o n - s i n g u l a r Borel G-set s o f p o s i t i v e u(r(s))> 3.28.

u-measure, t h a t i s ,

such t h a t

O.

Examples : (a)

T r a n s f o r m a t i o n group. Let u be a measure on the u n i t space U o f the t r a n s -

f o r m a t i o n group (U,S). A Borel subset o f U x S o f p o s i t i v e is a left and

u-measure

l - m e a s u r e , where I

Haar measure f o r S, c o n t a i n s a r e c t a n g l e A × B w i t h A,B Borel~u(A) > 0

I ( B ) > O. Choose s ~ B. Then s = { ( u , s )

of positive (b)

u ×

: u c A}

i s a n o n - s i n g u l a r Borel G-set

u-measure.

r-discrete

countable r-discrete

g r o u p o i d s . Let u be a measure on the u n i t space o f a second g r o u p o i d G. Let E be a Borel set in G o f p o s i t i v e

v-measure.

Since G can be covered by c o u n t a b l y many open G - s e t s , t h e r e e x i s t s an open G-set t such t h a t s = E of positive (c)

t has p o s i t i v e

~-measure. Then, s i s a n o n - s i n g u l a r Borel G-set

u-measure.

Transitive

the t r a n s i t i v e

principal

g r o u p o i d s . Let × be a l o c a l l y

compact space. We d e f i n e

g r o u p o i d on the space X as G = X x X, w i t h the g r o u p o i d s t r u c t u r e g i v e n

i n 1 . 2 . c and t h e p r o d u c t t o p o l o g y . We know t h a t a Haar system on G i s d e f i n e d by a measure ~ o f s u p p o r t ×. I f X i s u n c o u n t a b l e and s a t i s f i e s bility,

and i f

~ i s n o n - a t o m i c , then G has s u f f i c i e n t l y

This can be seen as f o l l o w s

t h e second axiom o f c o u n t a many n o n - s i n g u l a r Borel G-sets.

: t h e r e i s a Borel isomorphism o f X onto I n c a r r y i n g

i n t o the Lebesgue measure. Thus the problem is reduced t o the case X =I~ , = Lebesgue measure. Then the t r a n s i t i v e

g r o u p o i d i s isomorphic t o t h e g r o u p o i d

o f t h e t r a n s f o r m a t i o n group ( I R , IR) where IR a c t s by t r a n s l a t i o n conclude by a.

and we may

35 Question :

Assume t h a t G has s u f f i c i e n t l y

many n o n - s i n g u l a r Borel G-sets and t h a t

is a measure on GO q u a s i - i n v a r i a n t under every n o n - s i n g u l a r Borel G-sets ; can we conclude t h a t u is q u a s i - i n v a r i a n t ? The existence of s u f f i c i e n t l y

many n o n - s i n g u l a r Borel G-sets w i l l

be needed in

the second chapter (theorem 2 . 1 . 2 1 ) .

4. Continuous Cocycles and Skew-Products

The asymptotic range of a continuous one-cocycle ( d e f i n i t i o n 4.3) is used to solve a few problems concerning the t r i v i a l i t y

of cocycles and the i r r e d u c i b i l i t y

skew-products. This section c l o s e l y f o l l o w s [56],

of

[57] and [58] where a s i m i l a r study

has been done f o r C * - a l g e b r a s . Let G be a t o p o l o g i c a l groupoid ( d e f i n i t i o n space GO,

[El w i l l

2.1).

I f E is a subset o f the u n i t

denote i t s s a t u r a t i o n : [E] = r [ d - l ( E ) ] .

E is i n v a r i a n t (or i n v a r i a n t under G i f

I f E = [El, we say t h a t

there is any ambiguity). We w i l l

always assume

t h a t the range map r : G ÷ GO is open. Recall (2.4) t h a t l o c a l l y compact groupoids w i t h a left

Haar system have t h i s property. Then, the s a t u r a t i o n of an open subset of GO

is open. 4.1. D e f i n i t i o n : (i)

Let G be a t o p o l o g i c a l groupoid with open range map.

G is minimal i f

the only open i n v a r i a n t subsets of GO are the empty set

and GO i t s e l f . (ii)

G is i r r e d u c i b l e i f every non-empty i n v a r i a n t open subset of GO is dense.

I f there e x i s t s a dense o r b i t ,

then G is i r r e d u c i b l e . The converse holds i f G is

second countable and l o c a l l y compact. I t is useful to note t h a t the i r r e d u c i b i l i t y of G may be expressed as the density of the image of G in GO x GO by the map ( r , d ) G ÷ GO x

G~ x ÷ ( r ( x ) , d ( x ) ) .

These notions of m i n i m a l i t y a n d i r ~ d u c i b i l i t y

:

could

have been defined in terms of the s t r u c t u r e space GO//G of G, obtained from the q u o t i e n t space GO/G by i d e n t i f y i n g o r b i t s with the same c l o s u r e , but we w i l l

not make use of i t

36 here. The next p r o p o s i t i o n shows t h a t they are i n v a r i a n t under continuous s i m i l a r i t y . 4.2.

P r o p o s i t i o n : Suppose t h a t G and H are t o p o l o g i c a l groupoids which are continuous-

ly similar,

t h a t i s , which are s i m i l a r as in d e f i n i t i o n

1.3 where the homomorphisms

: G ÷ H and ~ : H ÷ G are continuous. Then the map 0 ÷ ( ~ 0 ) - I ( 0 ) sets up a b i j e c t i o n between the i n v a r i a n t open subsets of H and G. Proof :

Let 0 be an i n v a r i a n t open subset of H. The (@0)-1(0) is open and i n v a r i a n t .

For, if x ~ G and @Old(x)] c O, then @O[r(x)] Moreover,

(90° ¢0)-1(0) = 0

Indeed

(~ o@)(x) = (e o r ) ( x ) @0 o@O(u) = r [ 0 ( u ) ]

therefore

u ~ 0

iff

~0

~ 0 since 0 is i n v a r i a n t .

x (eod(x)) - I with

o

: r[#O(x)]

die(u)]

= u

#0 (u) c O. Q.E.D.

Let G be a t o p o l o g i c a l l i a n . We may s t i l l

groupoid and A a t o p o l o g i c a l group, not n e c e s s a r i l y abe-

define (cf 1.11) the f o l l o w i n g objects.

The set of continuous

homomorphisms from G to A is denoted by ZI(G,A). The subset of ZI(G,A)

consisting

of elements of the form c(x) = [ b o r ( x ) ] [ b o d ( x ) ] -1 where b is a continuous f u n c t i o n from GO to A is denoted by BI(G,A). Noreover, we say t h a t two elements c and c' in ZI(G,A) are cohomologous i f there e x i s t s a continuous f u n c t i o n b from GO to A such that c'(x)

:[bor(x)]

c(x) [bod(x)] -1.

The f o l l o w i n g d e f i n i t i o n version of the d e f i n i t i o n 4.3.

Definition

of the asymptotic range of a cocycle is the t o p o l o g i c a l

8.2 of [31,1].

: Let G be a t o p o l o g i c a l

groupoid, A a t o p o l o g i c a l group and c an

element of ZI(G,A). (i) (ii)

The range of c is R(c) = closure of c(G). The asymptotic range of c is R (c) = n R ( c u ) ,

taken over a l l

where the i n t e r s e c t i o n is

non-empty open subsets U of GO and c U denotes the r e s t r i c t i o n

of c to

GIU. Moreover, l e t u be a u n i t of G. (iii) (iv)

The range of c at u is RU(c) = closure of c(GU). The asymptotic range of c at u is Ru~ = n Ru (Cu), where the i n t e r s e c t i o n

37 is taken over a base of neighborhoods o f u. We use in the f o l l o w i n g d e f i n i t i o n A ; it

the character group A o f a t o p o l o g i c a l group

is the group o f continuous homomorphisms o f A i n t o the c i r c l e g r o u p T .

4.4. D e f i n i t i o n :

Let G be a t o p o l o g i c a l groupoid, A a t o p o l o g i c a l group and c an

element of ZI(G,A). The T-set of c is T(c) =

{x E A : xoC

E BI(G,~)}.

The f o l l o w i n g p r o p o s i t i o n gives some basic p r o p e r t i e s of the q u a n t i t i e s R (c) and T(c) ; in p a r t i c u l a r ,

they depend o n l y on the cohomology class o f c. The aim o f

t h i s section is to show t h e i r usefulness, j u s t i f y i n g

t h e i r i n t r o d u c t i o n . Further

references to the asymptotic range and the T-set o f a cocycle can be found in [31] in the context of ergodic theory. I t is i n t e r e s t i n g to note t h a t they were f i r s t duced on a work about operator algebras, namely, the Araki-Woods c l a s s i f i c a t i o n f a c t o r s obtained as i n f i n i t e 4.5.

Proposition :

introof

tensor products of f a c t o r s o f type I.

Let G be a t o p o l o g i c a l groupoid with open range map, A a t o p o l o -

g i c a l group and c ~ ZI(G,A). Then (i)

R (c) is a closed subgroup of A, T(c) is a subgroup of A, and R (c) and

T(c) are orthogonal to each o t h e r . (ii) (iii)

R (c) and T(c) depend o n l y on the class o f c. R (e) = {e} and T(e) = A, where e denotes the i d e n t i t y element of A as well

as the constant cocycle e(x) = e. Proof : (i)

Let us f i r s t

show t h a t R(c) R (c) c R(c). Suppose a c R ( c ) and b e R (c).

For every neighborhood V of b, r [ c - l ( v ) ] non-empty open subset 0 avoiding r [ c - l ( v ) ]

is dense in GO : i f not, there would e x i s t a and c o - l ( v ) would be empty. Let W be a

neighborhood o f ab and choose U,V open neighborhoods of a and b r e s p e c t i v e l y such t h a t UV c W. Since d [ c ' l ( u ) ]

is a non-empty open set and r [ c - l ( v ) ]

x, y ~ G such t h a t c(x) c U, c(y) E V and d(x) = r ( y ) .

is dense, there e x i s t

Then, c(xy) = c ( x ) c ( y ) ~

This shows a b e R(c). We deduce t h a t R (c) is s t a b l e under m u l t i p l i c a t i o n

UV cW.

: f o r any

non-empty open set U of GO, R (c) R (c) c R(Cu) R~(Cu) c R(Cu) hence R~(c) R (c) c R (c). As i t

is closed, symmetric and contains e, R~(c) is a closed subgroup o f A.

38 Since BI(G,T), with pointwise m u l t i p l i c a t i o n , We f i n a l l y

is a group, T(c) is a subgroup of A.

have to show t h a t f o r every x ~ T ( c )

and every a ~ R ( c ) , x(a) = 1. For

every closed neighborhood V of I in T, there e x i s t s a non-empty open set U in GO such t h a t (×oc)(Gu) c V because xoc ~ BI(G,T) ; in p a r t i c u l a r , x ( a ) ~ V. (ii)

Suppose t h a t c ' ( x ) = [ b o r ( x ) ] c ( x )

[bod(x)] -1 with c c ZI(G,A) and b a

continuous map from GO to A. Let a c RSc ). Vie want to show t h a t a ~ R ( c ' )

; that

i s , given a non-empty open set U' on GO and a neighborhood W' of a, we want to show t h a t W' n c'(GIu, ) # @.

We choose u E U ' ,

a neighborhood V of b(u) and a neighborhood

W o f a such t h a t VWV-1c W'. There e x i s t s an open neighborhood U o f u such t h a t b(U) cV.

Since W n c ( G i u ) # 9, we are done. We have shown R (c) c R ( c ' ) ,

R (c').

The e q u a l i t y T(c) = T(c')

(iii)

hence R (c) =

r e s u l t s from the d e f i n i t i o n o f a T-set.

Clear. Q.E.D.

S i m i l a r proofs y i e l d s i m i l a r r e s u l t s about the asymptotic range of a cocycle

at

a u n i t u. 4.6. P r o p o s i t i o n : Let G, A, c be as before and u ~ Go . Then (i) (ii) (iii) (iv) (v)

RU(c)

R~(c) : RU(c).

R~(c) is a closed subsemi-group o f A R~(c) depends only on the class R~(e) =

of c.

{e}

I f u ~ v, RU(c) = RV(c).

To proceed f u r t h e r , an a d d i t i o n a l assumption on the t o p o l o g i c a l groupoid G w i l l be needed. Let us r e c a l l the d e f i n i t i o n 4.7.

Definition :

3.24.i

;

Let G be a t o p o l o g i c a l groupoid. A G-set s ( d e f i n i t i o n

be c a l l e d a continuous G-set i f onto an open subset o f GO.

the r e s t r i c t i o n

1.10) w i l l

o f r and d to s is a homeomorphism

39

An open G-set o f an r - d i s c r e t e

locally

compact groupoid w i t h Haar measure i s a

continuous G-set. For a n o t h e r example, c o n s i d e r the groupoid o f a t o p o l o g i c a l

trans-

f o r m a t i o n group (U,S) ; l e t V be an open subset o f U and s c S ; then the G-set s = {(u,s)

: u c V}

is a continuous G-set.

In both examples, the groupoid admits a c o v e r

o f continuous G-sets. This is the assumption we need. 4.8.

P r o p o s i t i o n : Let G be a t o p o l o g i c a l

groupoid, A a topological

a b e l i a n group and

c c ZI(G,A). (i)

I f c c B I ( G , A ) , then f o r any neighborhood V o f e in A and any u c GO, t h e r e

e x i s t s an open neighborhood U o f u such t h a t R(Cu) c V. (ii)

I f G admits a c o v e r o f c o n t i n u o u s G - s e t s , i f

e x i s t s a dense o r b i t ,

GO i s compact and i f

there

then the converse h o l d s .

Proof : (i) (ii)

C l e a r since c ( x ) = b o r ( x ) - b o d ( x ) . We assume t h a t c s a t i s f i e s

the c o n d i t i o n t h a t f o r any neighborhood V o f e

i n A and any u c GO, t h e r e e x i s t s an open neighborhood o f u such t h a t c(Giu ) c V. This means i n p a r t i c u l a r introduce the principal lence r e l a t i o n We p r o v i d e i t

t h a t c vanishes on the i s o t r o p y group bundle o f G. L e t us g r o u p o i d a s s o c i a t e d w i t h G. I t

~ on GO. As a s e t , w i t h the f i n a l

it

is determined by the e q u i v a -

i s the image o f the map ( r , d )

t o p o l o g y , which i s u s u a l l y s t r i c t l y

: G -~ GO x GO .

finer

gy induced from GO x GO. The c o c y c l e c f a c t o r s through the map ( r , d ) c(x) = c'(r(x),d(x)).

V} i s an open c o v e r o f GO, t h e r e e x i s t s a f i n i t e

hence an entourageCLbof the u n i f o r m i t y

subcover

on GO such t h a t

cqJoand u ~ v ---->c'(u,v) c V.

Let us show t h a t c' Given ( u , v ) let

:

L e t V be a neighborhood o f 0 i n A. Since {U non-empty open set

i n GO such t h a t c'(UxU)

(u,v)

than t h e t o p o l o -

i s c o n t i n u o u s w i t h r e s o e c t t o the t o p o l o g y induced from GO x GO.

c GO x GO w i t h u m v ,

l e t x c G be such t h a t r ( x )

s be a continuous G-set c o n t a i n i n g x. Consider ( u ' , v ' )

sufficiently

c l o s e t o u, t h e r e e x i s t s y c s w i t h r ( y )

where w = d ( y ) ,

can be made a r b i t r a r i l y

= c'(w,v')

= u and d ( x ) = v and

w i t h u' m v ' .

= u' and c'

(u',w) - c'(u,v),

s m a l l . On the o t h e r hand c ' ( u ' , v ' )

can be made a r b i t r a r i l y

For u'

s m a l l , p r o v i d e d t h a t v'

- c'(u',w)

is c l o s e enough

40 to w, t h i s happens i f

u' is s u f f ~ c l e n t l ~ •

"

V

close to u and v' s u f f i c i e n t l y

c l o s e to v.

Next we show t h a t c' i s u n i f o r m l y continuous on the dense subset [Uo] x [Uo] o f ~0

x GO, where u 0 has a dense o r b i t .

c'(u',v')

- c'(u,v)

= c'(u',u)

I f u, v, u ' , v' are in the o r b i t o f u O, then

- c'(v',v).

T h e r e f o r e , c' extends t o a continuous

f u n c t i o n on GO x GO . Then, f ( u ) = c'(u,UO) i s a continuous f u n c t i o n on GO and agrees w i t h i t s coboundary on [u O]

c t

x [Uo],hence on G.

4.9. P r o p o s i t i o n : Let G be a t o p o l o g i c a l groupoid a d m i t t i n g a cover of continuous G-sets, A a t o p o l o g i c a l a b e l i a n group and c E Z I ( G , A ) . Suppose u 0 c GO has a dense orbit•

Then R~(c) =

R(Cu), where the i n t e r s e c t i o n is taken over a base o f neigh-

borhoods o f uO. Proof : Suppose t h a t a c R(Cu) f o r every U in a base of neighborhoods of uO. Let V,W be neighborhood o f e on A such t h a t W + W c V and U be a non-empty open set. There e x i s t s x ~ G w i t h r ( x ) = u O, d(x) c U and a continuous G-set s c o n t a i n i n g x. We may assume t h a t d(s) c U and c(s) - c(s) c W.Because a c R(Cr(s) ) , t h e r e e x i s t s y ~ G l r ( s ) such t h a t c(y) c a + !~• Let z = s - l y s , c(z) = c ( s - l r ( y ) )

+ c(y) + c(d(y)s)

~ a + W+

then z ~ Gld(s ) c GIU and Wc a + V•

Thus, a c R(Cu) f o r any non-empty open set U. Q•E •D. The f o l l o w i n g theorem may be compared

w i t h theorem 9 o f [ 3 1 , 1 ] . Combined w i t h

the r e s u l t s o f the second c h a p t e r , i t y i e l d s a p a r t i c u l a r

case o f a well-known theorem

o f Sakai which s t a t e s t h a t every bounded d e r i v a t i v e o f a simple C*-algebra w i t h identity 4.10.

is inner.

Theorem :

Let G be a t o p o l o g i c a l groupoid a d m i t t i n g a cover o f continuous

G-sets and a compact u n i t space, l e t A be a t o p o l o g i c a l a b e l i a n group and l e t c ~ Z I ( G , A ) . Assume t h a t G i s m i n i m a l . I f R(c) is compact and R (c) = { 0 } , then c c BI(G,A). Proof :

We use 4.8 ( i i ) .

Suppose t h a t there e x i s t s an open neighborhood V o f 0 in

A, u e G0, a base of neighborhoods of u and a net {x U} such t h a t

x u ~ GlU and

c(x u) ~ V.

41 I f {a U} is a subset of {C(Xu)} converging to a, then a # V and a E n R(Cu) where the i n t e r s e c t i o n is taken over a base of neighborhoods of u. By 4.9, a c R (c). Since R (c) = { 0 } , t h i s is a c o n t r a d i c t i o n .

Q.E.D.

Note t h a t i f A is t o r s i o n f r e e , the c o n d i t i o n R(c) compact already implies R (c) =

{0~.

The next theorem may be compared with th6or6me2.3.1 of [13] in the context of von Neumann algebras and w i t h theorem 4.2 o f [56] in the context of C~ - a l g e b r a s . The proof is adapted from [56]. 4.11. Theorem :

Let G be a t o p o l o g i c a l groupoid a d m i t t i n g a cover of continuous G-sets

and a compact u n i t space, l e t A be a l o c a l l y compact abelian group and l e t c c ZI(G,A). Assume t h a t G is minimal, then i f R(c)/ R (c) is compact in A/R ( c ) , i t f o l l o w s t h a t T(c) is the a n n i h i l a t o r o f R (c) in A. Lemma :

Let G be a t o p o l o g i c a l groupoid a d m i t t i n g a cover of continuous G-sets, l e t

A be a l o c a l l y compact a b e l i a n group and l e t c ~ ZI(G,A). Assume t h a t G is i r r e d u c i ble. Then iS= {V +R(cu) : V compact neighborhood of 0 in A and U non-empty open subsets of GO} is a base of a f i l t e r . Proof :

Its i n t e r s e c t i o n is R (c).

As in 3.4. of [56], i t s u f f i c e s to show t h a t given a compact neighborhood V

o f 0 in A and non-empty open s u b s e t s Ui o f G0, i = 1 , 2 ,

t h e r e e x i s t non-empty open

subsets Ui c Ui , i = 1,2 such t h a t R(Cu~)~ c V + R(Cu~ ) i , j

= 1,2 and i ~ j .

We choose

x c G with r ( x ) c U1 and d(x) e U2 and a c o n t i n u o u s G - s e t s c o n t a i n i n g x. We may

assume t h a t r ( s ) c UI, d(s) c U2 and c(s) - c(s) c V. Then U~ = r ( s ) and U½ = d(s) w i l l do.

Proof of the theorem : a base o f a f i l t e r

With the n o t a t i o n s of the lemma, the image o f ~

in A/R (c) is

of compact sets with i n t e r s e c t i o n {0}. Hence, given a neighborhood

V of 0 in A, we may f i n d a non-empty open set U in GO such t h a t R(Cu) c V + R (c). Thus, i f x is orthogonal to R ( c ) , R (×o c) = { I } . x

By 4.10,

e T(c). The reverse i n c l u s i o n has been shown in 4.5. ( i ) .

xoC ~ B I ( G , T ) ,

that is,

42 Recall t h a t , given a groupoid G, a group A and c ~ Z 1 (G,A), one may d e f i n e the skew-product G(c), whose u n d e r l y i n g space is G x A and u n i t space is GO x A. I f G and A are t o p o l o g i c a l and c continuous, G(c) w i t h the t o p o l o g y o f G x A i s a t o p o l o g i c a l groupoid. Note t h a t i f

G has an open range

map [resp. a cover o f continuous G - s e t s ] ,

then so has G(c).

The f o l l o w i n g c h a r a c t e r i z a t i o n o f the asymptotic range o f a cocycle i n terms o f the skew-product is taken from Pedersen [60] 8 . 1 1 . 8 .

It will

be used in Section 5 o f

Chapter 2. Recall t h a t t h e r e is a canonical a c t i o n o f A on the skew-product G(c), given by (x,b) • a = (x,a-lb)

4.12. P r o p o s i t i o n :

Let G be a t o p o l o g i c a l groupoid w i t h open range map, l e t A be

a t o p o l o g i c a l a b e l i a n group and l e t c ~ Z I ( G , A ) . Then the f o l l o w i n g p r o p e r t i e s are equivalent for a e A : (i)

a ~ R (c) and

(ii)

f o r any non-empty open i n v a r i a n t subset 0 o f the u n i t space G(c), 0 n 0 - a

is non-empty.

Proof : (i)

~

(ii)

Suppose a E R ( c ) .

Let 0 be a non-empty i n v a r i a n t subset o f GO × A. I t contains a non-empty r e c t a n g l e U × V, w i t h U open on GO and V open in A. Let b ~ V. Since a c R ( c ) , t h e r e e x i s t s x c GIU such t h a t c ( x ) E a - b + V. Then ( r ( x ) , b ) U x V n 0. Since Since ( r ( x ) , (ii)

~,

(r(x),

b - a) is e q u i v a l e n t to ( d ( x ) , b - a + c ( x ) ) ,

b - a) = ( r ( x ) , (i)

and ( d ( x ) , b - a + c ( x ) ) belong to

b) . a, i t also belongs to 0 . a

Suppose t h a t a s a t i s f i e s

it

belongs to 0.

.

(ii).

Let U be a non-empty open s e t i n GO and V be a neighborhood of 0 in A. Choose a neighborhood N o f 0 such t h a t W - W c V. Since the s a t u r a t i o n of U x W in the u n i t space o f G(c) is an i n v a r i a n t open s e t , i t contains an element ( v , b ) t o g e t h e r w i t h ( v , b - a ) . This i m p l i e s the e x i s t e n c e o f x and y i n G such t h a t r ( x ) = v and ( d ( x ) , b + c ( x ) ) e U x W and

43

r(y)

:

v and ( d ( y ) ,

b - a + c(y))

c U x W.

Then, x - l y ~ GIU and c ( x - l y ) = -c(x) + c(y) c a + W - W c a + V. This shows t h a t a e R (c). Q.E.D.

4.13. P r o p o s i t i o n :

Let G be a t o p o l o g i c a l groupoid with open range map, l e t A be

a t o p o l o g i c a l group and l e t c ~ ZI(G,A). The f o l l o w i n g p r o p e r t i e s are e q u i v a l e n t : (i)

G is i r r e d u c i b l e and R (c) = A and

(ii)

G(c) is i r r e d u c i b l e .

Proof : (i) ~>

(ii)

I t s u f f i c e s to show t h a t , given non-empty open sets U1,U 2 in GO,

a neighborhood V of e in A and a ~ A, there e x i s t s z ~ G such t h a t r ( z ) and c(z)

~ U1, d ( z ) c U2

c aV. Choose W, open neighborhood of e such t h a t W-Iw c V. Since G is

i r r e d u c i b l e , there e x i s t s b ~ A such t h a t c-l(bw) n r-1(U1 ) n d - l ( u 2 ) # @. Let U = r[c-l(bw) N r-l(Ul) t h a t c(x) d(y)

m bWa-1. Since r ( x )

~ U2 and c(y)

~>

(i)

Since ba -1 ~ R (c), there e x i s t s x ~ GIU such

E U, there e x i s t s y ~ G such t h a t r ( y ) = r ( x ) ,

e bW. Let z = x - l y . Then r ( z ) = d(x) ~ U c U1, d(z) = d(y) c U2

and c(z) = c ( x ) - l c ( y ) (ii)

n d-l(u2)].

c a W-1W c aV.

I f G(c) is i r r e d u c i b l e , then G is c l e a r l y i r r e d u c i b l e . To show

t h a t R (x) = A, l e t a E A, l e t V and W be neighborhoods o f e in A such t h a t W'Iw c V and l e t U be a non-empty open subset of G0. Since G(c) is i r r e d u c i b l e , there e x i s t s x ~ GU and b ~ W such t h a t bc(x) ~ Wa. Then c(x)

~W-1Wac Va. This shows t h a t

a ~ R(c). Q.E.D.

4.14.

P r o p o s i t i o n : Let G be a t o p o l o g i c a l groupoid with open range map, l e t A be a

t o p o l o g i c a l group and l e t c ~ ZI(G,A). Let (u,a) ~ GO × A.

(i) c at u, (ii)

I f (u~a) has a dense o r b i t r e l a t i v e to G(c), then the asymptotic range of RE(c ) , is equal to A. Conversely, i f G is minimal and i f R~(c) = A, then (u,a) has a dense o r b i t .

44

Proof : (i)

Suppose t h a t the o r b i t

GO x A.

[(u,a)]

= {(d(x),ac(x))

: x ~ Gu}

is dense in

Let b ~ A, l e t V be a neighborhood of b and l e t U be an open neighborhood

o f u.There e x i s t s x e Gu such t h a t ( d ( x ) , a c ( x ) ) c U × aV. Thus, x e Gu n GIU and c ( x ) e V. We conclude t h a t b c R~(c). (ii)

Suppose t h a t R~(c) = A. Let F be the c l o s u r e o f the o r b i t o f ( u , a ) . For

any b e A, ( u , b ) c F : indeed, l e t U be an open neighborhood o f u and V a neighborhood of b ; since a - l b ~ R~(c), t h e r e e x i s t s x such t h a t r ( x ) = u, d(x) E U and c(x) m a - l v

; in o t h e r words, ( d ( x ) , a c ( x ) )

E U × V. The set {v c GO : f o r any b e A,

( v , b ) e F} is non-empty, G - i n v a r i a n t and closed. Since G is m i n i m a l , t h i s i s GO , hence F = GO × A. Q.E.D. 4.15. P r o p o s i t i o n :

Let G be a t o p o l o g i c a l groupoid w i t h open range map, A a t o p o l o -

g i c a l group and c ~ Z I ( G , A ) . Assume t h a t A is compact,then R (c) = P,U(c) f o r every u e GO w i t h a dense o r b i t . Proof : I#e f i r s t

show t h a t R(c) = RU(c)-iRU(c) f o r u w i t h a dense o r b i t .

sion RU(c) -1 RU(c) c R(c) holds f o r a r b i t r a r y a dense o r b i t .

Since A i s compact, i t

o f RU(c) -1 RU(c).

The i n c l u -

u. Suppose now t h a t a c R(c) and u has

s u f f i c e s to show t h a t a belongs to the c l o s u r e

I f V is a neighborhood o f a, r [ c - l ( v ) ]

n [u] is non-empty : t h e r e

e x i s t x , y such t h a t c(x) E V, r ( x ) = d(y) and r ( y ) = u. Then, c(y) - I c(yx) ~ [c(GU) " I c(GU)] n V. Therefore R(Cu) = RU(cu ) - I RU(cu ) f o r any open neighborhood U o f u. Using the compactness o f A, one may w r i t e : R (c) = n R ( c u ) =

[nRU(cu)] -1

[nRU(cu )] = Ru (c)

where the i n t e r s e c t i o n s are taken over a l l

-1 RU(c) : R (c)

open neighborhoods o f u. The l a s t

e q u a l i t y holds because, in a compact group, any closed semi-group is a group. Q.E.D. 4.16.

Corollary :

Let G be a t o p o l o g i c a l groupoid w i t h open range map, l e t A be a

t o p o l o g i c a l group and l e t c m z l ( G , A ) . (i) (ii)

I f G(c) i s m i n i m a l , then G is minimal and R (c) = A I f A is compact, i f

G is m i n i m a l , and i f

R (c) = A, then G(c) is m i n i m a l .

45

Proof :

(i) (ii)

I f G(c) is minimal, G is c l e a r l y minimal. Moreover, R (c) = A by 4.12. Using 4.14 and 4.13 ( i i ) , o n e sees t h a t every (u,a) ~ GO × A has a dense

orbit.

Q.E.D.

4.17. P r o p o s i t i o n :

LetG be a t o p o l o g i c a l groupoid with open map, l e t A be a group

71(G,A). The follow,ring properties are with the d i s c r e t e topology and l e t c ~ ~ equivalent : (i) (ii)

G is i r r e d u c i b l e and R (c) = R(c) ; and c-l(e)

is i r r e d u c i b l e .

Proof : ( i ) ----> ( i i )

Let U1 and U2 be non-empty open sets in GO. By i r r e d u c i b i l i t y

of

G, there is a cA such t h a t c-1(a) n r-1(U 1) n d - l ( u 2 ) is non-empty. Then U = r~'1(a)

n r-l(u1)

n d - l ( u 2 )]

is a non-empty open set and since a "1 E R J c ) ,

there e x i s t s x ~ GU with c(x) = a -1. Therefore, there is y e G such t h a t d(x) = r ( y ) , c(y) = a and d(y) ~ U2. Consider z = xy : d(z) ~ U2, r ( z ) = r ( x )

c U1 and

c(z) = c ( x ) c ( y ) = e. This shows t h a t the groupoid c-1(e) is i r r e d u c i b l e . (ii)

~>

(i)

If c-l(e)

is i r r e d u c i b l e , so is G.Consider a ~ R(c) and U a non-

empty open subset of GO. Since c ' 1 ( e ) is i r r e d u c i b l e , c - l ( e ] n r - l ( u ) = V is a non-empty open set and so is c - l ( e )

nr-l[d(V)]

n d-l[r(c-l(a))]

n d-1(U). Therefore, we can

f i n d x, y, z such t h a t : c(x) = e, c(y) = a, c(z) = e, d(x) = r ( y ) ,

r ( z ) = d(y)

,

r ( x ) m U and d(z) ~ U. Then, xyz E GIU and c(xyz) = a. This shows t h a t a ~ R (c). Q.E.D. Another subgroup of A can be attached to a cocycle c e ZI(G,A) (cf. theorem of Section 2). We conclude t h i s section by discussing b r i e f l y r e l a t e d to R (c) and T(c) ± 4.18. D e f i n i t i o n :

[62],

how i t

is

(the a n n i h i l a t o r o f T(c) in A) in a p a r t i c u l a r case.

Let G be a t o p o l o g i c a l groupoid, l e t A be a t o p o l o g i c a l group

and l e t c c ZI(G,A). We define R1(c ) to be the set of elements a of A with the property t h a t f o r every G ( c ) - i n v a r i a n t complex-valued continuous f u n c t i o n on GO × A

46

GO × A, the e q u a l i t y f ( u , b a )

and f o r every ( u , b ) in 4.19.

Proposition :

R (c)

holds.

Let G be a t o p o l o g i c a l g r o u p o i d , l e t A be a t o p o l o g i c a l group

and l e t c ~ ZI(G,A). Then R ( c ) c R 1 ( c )

Proof :

= f(u,b)

c

T(c ±

c Rl(C ). I f a ~ R l ( C ) , there is a continuous f u n c t i o n f on GO × A,

which i s G(c) i n v a r i a n t ,

and ( u , b ) c GO × A such t h a t f ( u , b a ) # f ( u , b ) ,

hence there

e x i s t an open neighborhood U o f u and a neighborhood V of a such t h a t f(U × bV) n f(U x bVa - I )

= ~.If

x E GU and c(x) c V, then f ( r ( x ) , b ) = f ( d ( x ) , b c ( x ) ) .

This is a

c o n t r a d i c t i o n and t h e r e f o r e a ~ R ( c ) . RI(C)

c T(c) ±. Let a E RI(C ) and X ~ T ( c ) , t h a t is

e x i s t s g : GO ÷#

continuous such t h a t god(x)

f(u,b)

= g(u)

f(u,b)

t h a t i s , g ( u ) x ( b ) × ( a ) = g ( u ) x ( b ) , hence

xoc ~ B I ( G , ~ ) , Then, there

×oc(x) = g o r ( x ) f o r every x e G. Let

x ( b ) . Then f i s continuous and G ( c ) - i n v a r i a n t .

Therefore, f(u,ba) =

x ( a ) = 1. Q.E.D.

More i n f o r m a t i o n can be obtained in the case of a compact a b e l i a n group A. 4.20.

Proposition :

Let G be a t o p o l o g i c a l g r o u p o i d , l e t A be a t o p o l o g i c a l group

and l e t c e Z I ( G , A ) . Assume t h a t G i s minimal and A is compact and abeliano Then

RL(C) =

T(c) ~.

Proof : Let f be a continuous G(c)-invariant function on @0 #A. For each X E A, g(u) = f f ( u , a ) god(x)

x ( a ) d a is continuous and s a t i s f i e s

Xoc(x) = g o t ( x )

Since G i s m i n i m a l , e i t h e r g vanishes case, XOC ~ B I ( G , ~ ), t h a t i s , x f(u,-)

identically

o r not a t a l l

and, in the l a t t e r

~ T ( c ) . Thus, f o r every u, the F o u r i e r transform of

is supported on T ( c ) . Hence, i f a c T(c) ~, then f o r any b c A f(u,

a + b) = f ( u , b ) .

So a c R1(c ). Q.E.D.

We also r e c a l l

t h a t under the hypotheses o f 4.11, R (c) = R1(c ) = T(c) i . These

l a s t f a c t s , combined w i t h 4.15, give a theorem o f Rauzy ( [ 6 2 ] , theorem o f s e c t i o n 2) about the m i n i m a l i t y

of a skew-product.

CHAPTER I I THE C* -ALGEBRA OF A GROUPOID

First,

l e t us say that "groupoid" stands

f o r l o c a l l y compact groupoid

with a f i x e d Haar system ( d e f i n i t i o n 1.2.2) chosen once f o r a l l . We shall see ( c o r o l l a r y 2.11) how the C*-algebra can be affected by another choice of Haar system. We also assume that the topology of the groupoid is second countable. The goal here is to construct the C * - a l g e b r a of a groupoid in a way which extends the well-known cases of a group (e.g. Dixmier [19]) or of a transformation group (e.g. Effros-Hahn [23]). In f a c t , our construction closely fellows [23]: the space Cc(G) of continuous functions with compact support is made into a * -algebra and endowed with the smallest C*-norm making i t s representations continuous ; C*(G) is i t s completion. The d e t a i l s are in Section 1. We r e f r a i n from putting any modular function in the d e f i n i t i o n of the i n v o l u t i o n , since none is a v a i l a b l e . However, this is a minor change and the C * - a l g e b r a so obtained is isomorphic to the usual one in the case of a transformation group. Let us note t h a t , in the case of a transformation group, the * - a l g e b r a Cc(G) has been studied by Dixmier ( [16],§ X) in the context of quasi-unitary algebras. I f a is a continuous 2-cocycle on G with values in the c i r c l e group, the C * - a l g e b r a C*(G,~) is defined in the same fashion. One of the main j u s t i f i c a t i o n s f o r i t s i n t r o d u c t i o n , besides the need to deal with projective representations, is given in Section 4, where the C * - a l g e b r a of an r - d i s c r e t e p r i n c i p a l groupoid is characterized, under s u i t a b l e conditions, by the existence of a p a r t i c u l a r l y nice kind of maximal abelian subalgebra. One of these conditions is amenability, which is defined in Section 3.

48 An essential tool in the study of the C * - a l g e b r a of a groupoid is the correspondence, very f a m i l i a r

in the case of a group, between the u n i t a r y representations

of the groupoid and the non-degenerate representations of the C * - a l g e b r a .

I t is

established at the end of Section 1 and under a c o n d i t i o n (existence of s u f f i c i e n t l y many n o n - s i n g u l a r G-sets) s u f f i c i e n t Of p a r t i c u l a r

f o r our a p p l i c a t i o n s .

i n t e r e s t are the r e g u l a r representations of a groupoid. They have

been studied e x t e n s i v e l y since Murray and Von Neumann and we r e f e r to Hahn [45] for further details.

They appear under various forms, one of them is as representations

induced from the u n i t space ; t h i s is described in Section 2, where the inducing process from more general subgroupoids is also considered. The l a s t s e c t i o n , Section 5, i n t e r p r e t s r e s u l t s of Section 4 of the f i r s t tivity

and s i m p l i c i t y

in the language of C * - a l g e b r a s

the

chapter. They center around the question of p r i m i -

of a crossed-product algebra.

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

Let G be a l o c a l l y compact groupoid with l e f t

Haar system {~u} and l e t ~ be a

continuous 2-cocycle in Z2(G,T ). For f and g c Cc(G ), l e t us define f*g(x) f*(x)

= I f ( x y ) g ( y -1) ~ ( x y , y - 1 ) d ~ d ( X ) ( y )

,

= f ( x -1) o ( x , x - l ) .

1.1. Proposition : Under these operations, Cc(G ) becomes a t o p o l o g i c a l ~ - a l g e b r a , denoted by Cc(G,~ ). Proof : We f i r s t

show t h a t these operations are well

defined. For each x, f . g ( x )

is the value of the i n t e g r a l of a continuous f u n c t i o n w i t h compact support. Since f*g(x)

is nonzero only i f there is y such t h a t f ( x y ) and g(y-1) are nonzero,

supp(f.g)

is contained in the compact set (suppf)(suppg). To show the c o n t i n u i t y of

f ~ g, we may use the same device as Connes in [14] 2.2. That i s , since G2 is a closed subset of the normal space G x G, the f u n c t i o n ( x , y ) -~ f ( x y ) ~ ( x y , y -1) may be extended to a bounded continuous f u n c t i o n k on G x G. Since the f u n c t i o n

49 x ~ ~ : G ~ Cc(G), where ~(y) = k(x,y)g(y-1), (x,u)

is continuous, so is the function

÷)k(x,y)g(y-1)d~U(y)

: G x GO ÷C ; in particular,

(x,d(x)) is continuous. Note that f * =

its restriction to

is also continuous, with compact support suppf*

(suppf) -1. The convolution is associative : i f f, g, h ~ Cc(G), f * (g . h) (x) = !f(xy) g h(y -1) o(xy,y-1)d~d(X)(y) .lif(xy ) g(y-iz) h(z -1) ~(y-lz,z-1)~(xy,y-1)d~r(Y)(z)d~d(X)(y)

= ))f(xy) g(y-lz) h(z -I) {(xy,y -1) ~(y-lz,z-1)d~r(Z)(Y)dxd(X)(z)

= i)f(xzy) g(y-1) h(z-1) {(xzy,y-lz-1)o(y-l,z-1)d~d(Z)(y)d~d(X)(z)

= ))f(xzy

The involution f**(x)

g(y-1) h(z-l) ~(xzy,y-1) ~(xz,z-1)d~d(Z)(y)d~d(X)(z)

=

(f*g

(xz) h(z -I) u(xz,z-1)dxd(X)(z)

=

(f.g

. h (x).

s involutive

:

= f*(x -I) ~(x,x -I

= f(x) ~(x-l,x) ~(x,x -I) = f(x).

Also (f * g ) * ( x )

= f . g ( x -1) ~(x,x -1) = .(f(x-ly) g(y-1) ~(x-ly,y-1) ~(x,x-1)d~r(X)(y).

Using

~(x-ly, y-l) = ~(y,y-1) c~(x-l,y)

and

~(x, x-l) = ~(x-l,Y) ~(x-ly,y -Ix) o(y,y']x), we obtain (f.g)*(x)

=ig(y-l)

{ ( y , y - ! ) f(x-ly) ~(x-ly,y-lx) ~(y,y'lx)d~r(X)(y)

=Ig*(Y)

f*(y-lx)

=

g*

* f*(x)

~(Y'y-lx)d>r(X)(Y)

.

Finally, the operations are continuous. If fn ~ f and gm -~ g' there exist compact sets K and L such that, eventually, supp f n cK and sump gm c L. Then, supp fn*gm c KL. Also, If * g(x) - fn * gm(x)! ~-l]f(xy)g(y-1) - fn(xy)gm (y-1)]d~d(x)(y) _<.l[f(xy) - fn(XY) iIg(y-i)Id~d(X)(y) +lIfn(XY ) IIg(y -I) -gm(y -I) Id~,d(X)(y)

50 Therefore, f n * g m converge uniformly to f , g on KL. Moreover, supp fn a K-1 and Ifn(X) - f * ( x ) I

= Ifn(X -1) - f ( x - 1 ) I

converges uniformy to zero on K- I .

q.E.D. 1.2. Proposition : I f a and ~' are cohomologous, then Cc(G,~ ) and Cc(G,~' ) are isomorphic. Proof : I f o ' ( x , y ) = a ( x , y ) c ( y ) c ( x y ) - l c ( x ) Cc(G,~' ) to Cc(G,d ) which sends f to fc.

we can define the isomorphism ¢ from Indeed f o r f , g s Cc(G)

¢(f) . ¢(g)(x) = j f ( x y ) c ( x y ) g ( y - I ) = .I f ( x y ) g ( Y -1)

o(xy,y-1)dxd(X)(y)

o'(xy,y'l)c(x)dxd(X)(y)

¢ ( f ~ g)(x) ¢ ( f ) * (x) = ¢ ( f ) ( x ' l i =

f(x-li

~(x,x - I )

= f(x -I) c(x -I) o(x,x -1)

~ ' ( x , x -1) c(x) = # ( f * ) ( x ) . Q.E.D.

1.3. D e f i n i t i o n

: A representation of C~(G_,o) on a H i l b e r t space H is a *-homomor-

phism L : Cc(G,o ) ÷ ~ ( H ) which is continuous when Cc(G,~ ) has the inductive l i m i t topology and~5(H) the weak operator topology, and is such that the l i n e a r span of {L(f)(

, f c Cc(G,o),~ c H} is dense in H.

The I-norm introduced by P. Hahn in

~ 5 ] , page 38, w i l l

be a convenient estimate

f o r the C*-norm we wish to define on Cc(G,a ). I t is worthwhile to notice t h a t t h i s norm is used by numerical analysts, in the case when G is the t r i v i a l

equivalence

r e l a t i o n on a set of n elements (e.g. in i n t e r p o l a t i o n theory). Let us r e c a l l i t s d e f i n i t i o n - or r a t h e r , the d e f i n i t i o n appropriate to our s e t t i n g . For f ~ Cc(G ) ] I f l l l , r = sup j - I f l d ~ u, u~G0

IIfill, d = sup u~G0

I Ifld~ u ; and [Ifll I = max( IIflll ,r, l l f I I l , d ) -

1.4. Proposition : (i) topology.

If. IfI is a norm on Cc(G ) defining a topology coarser than the inductive l i m i t

51

(ii)

For any oBZ2(G,T),

II ITI is a * - a l g e b r a norm on Cc(G,~ ).

Proof : ( i ) I t suffices to look at IIfIIi, r. I t is routine to check that i t is a norm. Suppose that fn ÷ 0 in Cc(G)" Then' because of the c o n t i n u i t y of the map : Cc(G) ÷ Cc(GO) which sends f to Ifd~ u, i t follows that Cc(GO) and a f o r t i o r i infinity, (ii)

~(IfnT)

tends to zero in

in the space Co(GU) of continuous functions tending to 0 at

equipped with the supnorm. To show that Ill , gll I < llfll I IIgIII, i t s u f f i c e s to consider IT Ill, r. Then

f o r f,g c Cc(G), l ' I f * gl d~u < i . I I f ( y ) I

Ig(y-mx)Id~r(X)(y)d~U(x),

<

IIIf(y)llIg(y-lx)Id~r(y)(x)

<

ilf(Y)[

(becauseI~ I =1)

d~U(y)

l ' I g ( x ) I d ~ d ( Y ) ( x ) d~U(Y)

< sup ~Ig(x)Id~V(x) x i l f ( y ) I d ~ U ( Y ) < I l g I I i , r l I f l i l , rF i n a l l y , by d e f i n i t i o n ,

llf~l I

= llfll I . Q.E.D.

1.5. D e f i n i t i o n : A representation L of Cc(G,~ ) w i l l ltL(f)tE

lIrllz

be called bounded i f

f o r a l l f ~ Cc(G,~ ).

We may define

Ilfll

= sup IJL(f)tl where L ranges over a l l bounded representations

of Cc(G,~ ), and make two comments. ( i ) C l e a r l y I1-11 is a c ~ -semi-norm. I t w i l l sufficiently (ii)

be shown soon, by e x h i b i t i n g

many bounded representations, that i t is a norm.

For a large class of groupoids (including transformation groups), we w i l l

e s t a b l i s h in Corollary 1.22 that every representation on a separable H i l b e r t space is bounded. This is done in a fashion s i m i l a r to [24], 4.9, page 45 in the case of a transformation group. The notion of H i l b e r t bundle (or more p r e c i s e l y H i l b e r t space bundle) used in the next d e f i n i t i o n is given in [61]page 264. The base space of such a bundle is a standard measure space and each f i b e r is a separable H i l b e r t space.

52 1.6. D e f i n i t i o n :

Let o be a (continuous) 2-cocycle. A o - r e p r e s e n t a t i o n o f G consists

of a q u a s i - i n v a r i a n t measure ~ on GO and a ( o , G ) - H i l b e r t b u n d l e % o v e r (GO,~). More p r e c i s e l y , there is a map L : G + Iso(J~) = { i s o m e t r i e s #u,v :J~v

-~

where u, v ~ GO}

such t h a t ( i ) L(x) sends J~d(x) onto JCr(x) For x e G and L(u) = idJ~0

f o r u ~ GO "

0

(ii)

L(x)L(y) =

~ ( x , y ) L(xy) f o r v~ a.e. ( x , y ) , where v 2 is the induced measure

on G2 ; (iii)

L(x) -1 = ~

( i v ) x ~ (L(x)

f o r ~ a.e.

x, where v is the induced measure on G ; and

~od(x), ~or(x)) is measurable f o r ever'y p a i r of measurable

sections ~ and n. Two ~-representations (u,JC,L) and (~',JC',L') are e q u i v a l e n t i f

the measures

and u' are e q u i v a l e n t and there e x i s t s an isomorphism ~ of J C o n t o J £ ' ( i n the sense o f [~1]) which i n t e r t w i n e s L and L ' , t h a t i s , such t h a t L ' ( x ) ~.d(x) = ~or(x) L(x) f o r va.e.x. Let ( u , j ~ , L )

be a ~ - r e p r e s e n t a t i o n o f G. Then r (J~), or r(J~) when there is no

ambiguity about ~, denotes the H i l b e r t space of s q u a r e - i n t e g r a b l e sections w i t h respect to u. The modular f u n c t i o n of ~ is denoted by D and i t s symmetric induced measure is ~0 = D-I/2v 1.7. P r o p o s i t i o n :

(see 1.3.4).

Let (p,J~,L) be of a

o - r e p r e s e n t a t i o n o f G.For

g,n c r (J~) and

f e Cc(G), set (,)

(k(f)~,q)

(i)

This defines a bounded r e p r e s e n t a t i o n of Cc(G,o ) on

(ii)

= !f(x)

(L(x) ~od(x), n o r ( x ) ) dvo(X ). r(JC).

Two e q u i v a l e n t o - r e p r e s e n t a t i o n s of G give two e q u i v a l e n t representations

of Cc(G,o ) . Before s t a r t i n g the proof, l e t us make two remarks. a.

Let Up = {u c GO : dim J£u = p}

subset. Let

Up be the r e s t r i c t i o n

Then ( p p , % , L ) onr~(J£)

is a

f o r p = 1,2 . . . . .

~. I t is an i n v a r i a n t measurable

of u to Up.

o - r e p r e s e n t a t i o n of G, which defines by (*) an o p e r a t o r L p ( f )

; the operator L ( f )

is the d i r e c t sum o f the L p ( f ) ' s .

Therefore, i t

is

53 s u f f i c i e n t to consider the case when dim J~u is constant. Then JC,is isomorphic to a constant Hilbert bundle with f i b e r K and ? (J~) is isomorphic to the space L2(GO,u,K) of square-integrable K-valued functions on (GO,u). Moreover, for f E Cc(G) and ~ L2(GO,~,K), L(f)

is given by

L(f)~ (u) = ~ f ( x ) k(x)

~od(x) D-1/2(x)

d~U(x) ~ a.e.

where the r i g h t hand-side may be interpreted as a weak integral in K. b.

In the case of a group, GO is reduced to one element and there is a unique inva-

r i a n t measure class. Therefore, there is no need to mention i t . Then, a ~-representation in the sense of 1.6 is a projective representation in the usual sense (e.g. [74] page I00). Assume that ~ = i. Then, the representation given by ( * )

is the integra-

ted form of the unitary representation L. I t is not the usual expression since our d e f i n i t i o n of the involution d i f f e r s from the usual one by the absence of the modular function. To get its usual expression, i t suffices to use the remark ( i i )

following

1.12. Proof of the proposition : By remark a, we may assume that J£is a constant Hilbert bundle and that (i)

Let us check that L(f) is a well-defined bounded operator. The map

x ~ f ( x ) (L(x) If(x)l ~0

=

F(jC=) = L2(GO,~,K).

~od(x), nor(x)) is measurable and dominated in absolute value by

l~od(x)I

Inor(x)l.

D-1/2v' - 1 ~ If(x)l

= D-I

This last function is vo-integrable, because

and

I~od(x)I

use of the Cauchy-Schwarz inequality yields

Inor(x)l

d~o(X)

_< [ I l f ( x ) l

l~od(x)l 2 du-Z(x)] I/2 [ j I f ( x ) l

< [!If(x)I

d~u(X)

1/2 -< Nfll I,d

I~I

_< IIfIIi I~I

I~(u)I 2 du(u)] 1/2 [ ~ I f ( x ) l

d ~ U ( x ) I n ( u ) l 2 du(u)] 1/2

I/2 IIfIll, r Inl

I~I

Therefore, x w f(x) (L(x) operator of norm

I~or(x)l 2 dr(x)] 1/2

~od(x), nor(x)) is vo-integrable and L(f) is a bounded

IL(f)II _< IIfII1- The continuity

of the map L : Cc(G) ÷~(r(J~)) follows

from the previous line. We have to check that L is a .-homomorphism. So, let f,g be in Cc(G). On one hand, (L(f . g)~,q) is equal to ~( i f ( x y ) g(y-1) o(xy,y-1)dxd(X)(y)

(L(x) ~od(x), qor(x)) D1/2(x) dXu(X)d~(u)

54 = i f(xy) g(y-1)

~(xy,y-1) (L(x)

god(x), nor(x)) D1/2(x) d~2(x,y).

The use of Fubini's theorem is j u s t i f i e d since we are integrating l o c a l l y integrable functions on compact sets. On the other hand, we may write the equation L(g)g(u) : ~ g ( y )

k(y)

is equal to ~ f ( x ) g ( y ) Setting (x,y)

god(y) D-I/2(y) d~U(y) ~ a.e., so that

(L(x)(k(y)

(L(f)L(g)g,n)

god(y), nor(x)) D-1/2(y)d~d(X)(y) dXu(X)du(u ).

÷ (xy,y -1) and using a result in the proof of 1.3.3, we obtain

f(xy)g(y -1) a(xy,y -1) (L(x)

gor(y),

nor(x)) Dl/2(x) d~2(x,y). This shows that

L(f , g) = L(f)L(g). Next, for f s Cc(G), (L(f*)g,n)

= ~ f(x -1) a(x,x -1) (L(x)

~od(x), ~or(x)) d~o(X)

.f f(x) ~ ( x - l , x ) (L(x -1)

gor(x), nod(x)) duo(X),

(by symmetry of ~0). =

~f(x) a ( x - l , x ) a(x,x -1) (gor(x), L(x)nod(x)) d~o(X )

=.If(x) =

(gor(x), L(x)

nod(x)) d~o(X)

(g,L(f)n).

F i n a l l y , the representation L is non-degenerate. Indeed, let n be a vector of F ( ~ ) such that (L(f)g,n) = 0 for every f s Cc(G) and every g c r ( J £ ) . god(x), nor(x)) = 0 for ~ a . e . x .

Then, (L(x)

Choosing a countable total set in K, one sees that

nor(x) = 0 for ~ a.e.x.Hence n(u) = 0 for ~ a . e . u . (ii)

Let (u,Jg,L) and ( ~ ' , J g ' , L ' )

be two equivalent a-representations. Let g

be a positive l o c a l l y integrable function on (GO,u) such that u' = g~ and ~ an isomorphism of;}gontoJg' intertwining L and L'. For g s r by the formula g'(u) = g -1/2(u)

(jc~)

define

g' s r ,(jg')

#(u) g(u). Then, the map g ÷ gl is an isometry, also

denoted ~, of r (J~) onto r , ( j £ ' ) which intertwines the integrated representations L and L'. For, L(f)g(u) = ~f(x) L(x) L'(f)g'(u)

=~f(x)

god(x) D-i/2(x) d~U(x)

L'(x) g'od(x) D ' - i / 2 ( x ) d~U(x) and

D'(x) = gor(x) D(x) (god(x)) -1 v a.e. (1.3.3). Thus ( L ' ( f ) ~ g ) ( u ) ! f ( x ) L'(x)(god(x) - I / 2

~od(x)

is equal to

god(x)(gor(x)) -1/2 D-I/2(x)(god(x))I/2dxU(x)

55 : :

~ f ( x ) g(u) - I / 2 ¢(u) L(x) (@L(f)~)

~od(x) D-I/2(x) d~U(x)

(u).

Q.E.D.

I t is now easy to construct a f a i t h f u l family of bounded representations of Cc(G,o ), namely the regular representations.As in the case of a group, they play an essential role in the theory of groupoids. They have been defined and thoroughly studied by P. Hahn in [45], where i t is pointed out that they have been long-time f a v o r i t e s to produce von Neumann algebras (by the so-called group-measure space construction). Let ~ be a 2-cocycle and u a quasi-invariant measure. Consider the measurable f i e l d of H i l b e r t space {L2(G,~U), u E GO}

with square integrable sections

I m L2(G,~ u) du(u) = L2(G,v). For x E G, define L(x) mapping L2(G,~ d(x)) to L2(G,~r(x)) by L(x)~(y) = ~ ( x , x - l y ) ~ ( x - l y ) . This y i e l d s a G-representation of G (cf. example 3.11 of [45]) : (L(x)L(y)C)(z)

: ~(x,x-lz)(L(y)~)(x-lz)

~(x,x-lz)~(y,y-lx'Iz)~(y-lx-lz) ~(x,y)o(xy,y-lx-lz)~((xy)-lz) : o(x,y) L(xy)g(z). The argument 1.1 shows that the function (L(x)

god(x),

nor(x)) =

. I ~ ( x , x - l y ) ~ ( x - l y ) n(Y) d~r(X)(Y) is a continuous function of x for ~,n ~ Cc(G). Since any vector in L2(G,~) is a pointwise l i m i t of a sequence in Cc(G), this function is measurable when ~ and n are in L2(G,~). 1.8. D e f i n i t i o n : The above presentation of G on

c-representation of G w i l l be called the a-regular re-

~. I t s integrated form is the regular representation on ~ of

Cc(G,o)I t is a basic fact ([45], theorem 2.15) that the regular representation on ~ is the l e f t representation of a l e f t H i l b e r t algebra. We reproduce i t in our context. The main ingredient of the proof, which is the construction of a l e f t approximate

56 identity,

will

be needed in other places.

1.9. Proposition : The algebra Cc(G,o ) has a l e f t ductive l i m i t

approximate i d e n t i t y

topology).

Proof : Let us say t h a t a subset A of G is d - r e l a t i v e l y relatively

compact i f A n d - l ( K ) is

compact f o r any compact subset K of G0. Then, i f L is r e l a t i v e l y

AL = ( A n d - l ( r ( L ) ) ) L

is also r e l a t i v e l y

system of d - r e l a t i v e l y (Ki) a l o c a l l y f i n i t e relatively

( f o r the i n -

compact. Let us show t h a t GO has a fundamental

compact neighborhoods. Let V be an open neighborhood of GO and r 6 1 a t i v e l y compact open cover of GO ( i n G). There e x i s t s a

compact open set Ui in G such t h a t Ki c U i c V

n d-l(Ki).

an open neighborhood of GO contained in V and is d - r e l a t i v e l y any compact subset K of GO meets only a f i n i t e ned in a f i n i t e

compact,

number of K i ' s ,

Then U = u Ui is

compact. Indeed, since U n d-l(K)

is c o n t a i -

union of U i ' s . Let (Us) be such a fundamental system, w i t h Us c U1

f o r every s and l e t (Ks) be a net of compact subsets of GO increasing to G0. We can f i n d non-negative gs c non-negative h

~

Cc(G ), p o s i t i v e on Ks

and

Cc(G° ) such t h a t ha(u ) = (Ig~d~U) - I f o r u E Ks. Let us define

f (x) = hs or (x) g~(x). Then, f claim t h a t ( f s ) is a l e f t supp(f

and with support contained in Us

~ Cc(G ), supp f

approximate i d e n t i t y .

c U~ and ~ ( f s ) = 1 on Ks. We

Let f ~ Cc(G ) w i t h K = suppf. Then

. f ) and suppf are contained in the compact set L = UIK. I f ~ > 0 is given,

the using the compactness of L and the c o n t i n u i t y of f,~ f i n d s 0 such t h a t f o r ~ > s 0 and every ( x , y ) ~ L × U l{(y,y-lx)

and the product, one may

nG 2, I f ( y - l x )

- f(x)I

< ~ and

- II < E w h i l e r(L) c K s. I t follows t h a t

fs * f ( x ) - f ( x ) = + Ifm*f(x)

- f(x)l

lf

(y)[f(y-lx)

f(x)

- f(x)]~(y,y-Zx)d~r(X)(y)

• if (y)E~(y,y-lx)

~ c+ sup I f ( Y ) [ Y

- I]

d ~ r ( X ) ( y ) , and

~ f o r x c L. Q.E.D.

I f ( f s ) is a l e f t

approximate i d e n t i t y ,

(fs*)

is a r i g h t approximate i d e n t i t y .

I have not been able to prove the existence of a two-sided approximate i d e n t i t y f o r Cc(G ) except in p a r t i c u l a r cases ( r - d i s c r e t e groupoid and transformation groups). The d e f i n i t i o n

of a generalized H i l b e r t algebra, used in the next p r o p o s i t i o n ,

57 can be found in

[73] , pages 5 and 6.

1.10. Proposition :

(Cf. theorem 2.15 of [ 4 5 ] ) . Let ~ be a 2-cocycle and ~ a quasi-

invariant measure. Then (i) Cc(G,~ ) with the inner product of L2(G,~ -1) is a generalized Hilbert algebra ; and (ii)

i t s l e f t representation is equivalent to the regular representation on ~.

Proof :

Let us check the axioms of

[73].

(I) For f,g and h in Cc(G,~ ), (g, f * * h )

= Ig(y ) f * *

h(y) dv-l(y)

=J~g(y) f*(yx) h(x -1) o(yx,x -1) d~d(Y)(x) d~u(y ) d~(u) =JJJg(y) f*(yx) h(x -1) o(yx,x - I ) d~r(x)(y ) d~U(x) du(u) (meuse of Fubini's theorem is j u s t i f i e d because the function (x,y) ~ f*(yx)

f(y)

h(x-1)o(yx,x-1), defined on Gu x Gu, is continuous with compact support) =JJ/g(y) f*(yx - I ) h(x) o ( y x - l , x ) d~d(x)(Y) d~u(X) d~(u) =///f(y-1)

f*(y-Zx-1 ) h(x) o ( y - l x - l , x )

=//fg(y-1) f ( x y ) o ( y - l x - l , x y )

d~d(X)(Y) d~u(X) d~(u)

h(x) o(y-lx-l,x)d~d(X)(Y) d~u(X) d~{u)

: I I I f ( x y ) ~ y - 1 ) o(xy,y-1)d~d(X)(y) h(x) d~u(X ) d~(u) :

( ~ . g, h).

( I I ) For every f s Cc(G), g~ f . g is continuous. In fact, this operator has norm ~ Ilflli, as i t can be seen d i r e c t l y or deduced from ( i i ) .

(III)

Since Cc(~,~ ) has a l e f t approximate i d e n t i t y , the set {f * g : f, g ~ Cc(G)}

is dense in Cc(G) with the inductive l i m i t topology and a f o r t i o r i

with the L2(G,u - I )

topology. (IX) We have to show that the i n v o l u t i o n , as a real l i n e a r operator, is closable. Suppose that fn ÷ 0 and f *n ÷ g. Then J I f n l 2d~-1 ÷ 0 and f Ifn(X) * - g(x)I 2 d v - l ( x ) = -1 ~tfn(X) - g*(x)I 2 dr(x) ÷ O. Thus there is a subsequence fnk such that fnk÷ 0 v a.e. and f

nk

÷ g*va.e.

Since ~ and - 1 are equivalent, g*

= 0 va.e., hence g = O.

( i i ) Let us call L' the l e f t representation on L2(G,~-I), L ' ( f ) g = f , g, and L the regular representation on u acting on L2(~,v). The isometry from L2(G, ~) onto

58 L2(G,v -1) which sends ~ into ~' = D 1/2 ~ implements t h e i r equivalence. For ~,n

c L2(G,~) and f c Cc(G,v), (k'(f)C',

n') = - ~ f

, ~'(y)

= ISJf(x) = ~f(x)

n ' ( y ) dv-l(y)

~ ' ( x - l y ) ~(x,x-ZY) n'(y) d~r(Y)(x) d~u(y) d~(u) D1/2(x-ly)

~(x-!y)

~ ( x , x - l y ) D1/2(y) n(Y)

d~r(Y)(x) d~u(y ) d~(u) =J~f(x) =~f(x)

~(x-ly) {(x,x-ly)

O-I/2(x) d~r(Y)(x)d~U(y) du(u)

~(x-ly) { ( x , x - Z y ) n(y) d~r(X)(y) D-1/2(x) d~U(x) d~(u)

= ~ f(x) (L(x) ~od(x), =

n(y)

nor(x)) duo(X )

(L(f)¢,n). Q.E.D.

By looking at the polar decomposition of the i n v o l u t i o n , we obtain the usual ingredients of the Tomita-Takesaki theory : the modular involution J : L2(G,v - I ) ÷

L2(G,u -1) is given by J~(x) = Dl/2(x) C(x -1) ~ ( x , x - l ) , and the modular operator A

is defined on L2(G,v) n L2(G,u - I ) by A~(x) = D(x) 1.11.

Proposition : Cc(8,~ ) has a f a i t h f u l

~(x).

family of bounded representations, con-

s i s t i n g of regular representations. Proof :

Let u be a quasi-invariant measure with induced measure u and l e t L be the

regular representation of Cc(G,o ) on p. The kernel of L is {f ~ Cc(G,o ) : f vanishes on suppu}. For, i f f vanishes on suppu, the formula 1.7 (*) shows that L(f) = 0 ; while conversely, i f f * g = 0 in L2(G,u -1) for any g c Cc(G ), then using a r i g h t approximate i d e n t i t y , one sees that f = 0 in L2(G,~-I), so that f vanishes on suppl. To conclude, we observe that GO has a f a i t h f u l

family of quasi-invariant measures,

the t r a n s i t i v e measures ( d e f i n i t i o n 1.3.9). Q.E.D. 1.12. Definition : Let ~ be a 2-cocycle. The C*-algebra C~(G,~) is the completion of Cc(G,e ) for the C*-norm defined in 1.5. I t is called the o-C'algebra of the groupoid G. The C*-algebra of G is C*(G) = C*(G,1).

5g Remarks

:

( i ) I t results from 1.2 that cohomologous 2-cocycles give isomorphic C*-algebras. (ii)

In the case of a group or a transformation group, our d e f i n i t i o n does not

quite agree with the usual one (eg. [24] p. 35) because of the absence of a modular function in the involution. However, the Ce-algebras are isomorphic. To see t h i s , l e t G = U x S. We denote by Cc(G) the .-algebra of 1.1 and by Cc, (G) the of [24]. The involution for the l a t t e r is

f . (u,s) = f * ( u , s )

.-algebra

A(s-1), where ~ is

the modular function of the group S. Then the map from C~(G) to Cc.(G ) sending f to f'(u,s)

= f(u,s)A-I/2(s)

is a ~-isomorphism. I t extends to an isomorphism of

C*(G) onto C . ( G ) . (iii)

I f G is second countable, then Cc(G) with the inductive l i m i t topology is

separable, therefore C ~(G,o) is separable. I f h is a bounded continuous function on GO and hf(x) = hor(x) fh(x) = f(x)

f c Cc(G), we define

f ( x ) , and hod(x).

Then, hf and fh c Cc(G) and the following relations hold in the w-algebra Cc(G,o). For every f , g E Cc(G,o ), f * hg = fh * g, h(f . g )

= hf * g, and

(hf) * = f'h'where

h*(u) = h(u).

For example, f , hg(x) = Sf(xy) hg(y -1) ~(xy,y-1)d~d(X)(y) ~f(xy) hod(y) g ( y - l ) ffh(xy) g(y-l)

~(xy,y-l) d~d(X)(y)

~(xy,y-Z)

d~d(X)(y)

fh * g (x) In other words, h acts on Cc(G,~ ) as a double c e n t r a l i z e r (cf. [47]), Moreover i t acts continuously with respect to the inductive l i m i t topology. 1.13. Lemma :

I f L is a representation of Cc(G,a ), there exists a unique representation

60 M of Cc(GO) such that for every h ~Cc(GO ) and every f e Cc(G,o ), L(hf) = M(h)L(f) and L(fh) = k(f)M(h). Proof : Let H be the space of the representation L and H0 the l i n e a r span of n

{ L ( f ) ~ : f ~ Cc(G,~), ~ E H}. Let us try to define M(h) on H0 by M(h) (~ L ( f i ) ~ i ) n I = ~ L ( h f i ) ~ i , j u s t as in [47], page 317. To show that M(h) is well I defined, i t suffices to prove n

n

Z L ( f i ) ~ i : 0 =>Z L(hfi)C i : O. 1 1 Let ( f )

be a l e f t approximate i d e n t i t y

n

1

for Cc(G,o ). Then

n

L ( h f i ) ~ i = lim

n

Z L(h(f~)*fi))~ 1

i : lim

L(hf * fi)~i l

n

= lim L(hfa)

~ L(fi)~i

= 0 .

Moreover, M(h) satisfies (M(h)L(f)~,L(g)n)

: (L(hf)~,L(g)n)= = (¢,k(f*

(~,L(hf)*L(g)n)

* h*g)n) = ( ~ , k ( f * ) k ( h * g ) n )

= (k(f)~,k(h*g)n)

= (k(f)~,

M(h*)k(g)n).

To show that M(h) is a bounded operator, one uses as in [24], page 41, the r e l a t i o n (hg)

. (hf) +(kg)

. (kf) =

!!hII 2 g . f

valid for every h ~ Cc(G0), f, g E Cc(G,~), where k(u) = (llhll2 - lh(u)I2) I/2. Then n

IIM(h)

Z L ( f i ) ~ i N2 = Z 1 i,j

(L(hfi)~ i, L(hfj)~j) *

= .Z. (L((hfj

. (hfi))Ci,~j)

1,J =

llhll2

1,0X( L ( f j * ,

,

*fi)~i'~J)

-

i,j~ ( L ( fk j )

*

*

(kfi))~i,Ej)

=llhll 2 II~. k ( f i ) ~ i l l 2 - II ~ k ( k f i ) ~ i l l 2 I i -< Ilhll2 II

!L(fi)Cill 2" 1

Therefore M(h) extends to a bounded operator on H. It is then routine to check that H

is a representation of the

*-algebra Cc(GO) and that L(fh) = L(f)M(h).

Q.E.D.

61 1.14. Proposition : The C*-algebra C*(G O) is a subalgebra of the m u l t i p l i e r algebra of C *(G,a). Proof : The action of Cc(GO) as double centralizers of Cc(G,~ ) extends to C (G,o), because for every bounded representation L of Cc(G,~ ),

IIL(hf)ll

~ NM(h)II

Hhfl[ £

llhN ] l f l l .

llL(f)ll

~

Ilhli Hfll

, hence

This gives a*-homomorphism of C ~(G O) into the m u l t i p l i e r algebra of C*(G,~) which is v i s i b l y one-to-one. Q.E.D. The notion of generalized expectation used in the next proposition was introduced by M. Rieffel in [63] ( d e f i n i t i o n 4.12) in a context close to this one. We shall Iook at i t again in the second section. n

1.15. Proposition :

The r e s t r i c t i o n map Cc(G,a ) ÷ Cc(G~) is a generalized expectation.

I t i s smooth and f a i t h f u l .

The proof w i l l be given in a more general s i t u a t i o n in the second section (2.9). Remark : I f G is r - d i s c r e t e , C*(GO) is a subalgebra of C*(G,a) and the r e s t r i c t i o n map of Cc(G) onto Cc(GO) extends to an expectation of C*(G,~) onto

C~(GO). In this

case, C*(G,a) has a unit i f f GO is compact. I t w i l l be convenient in the following discussion to enlarge the class of functions on G. 1.16. Proposition : Let B(G) denote the set of bounded Borel functions on G with compact support. With convolution and involution defined as in 1.1, B(G) can be made into a * - a l g e b r a , denoted B(G,a). The proof is s i m i l a r to 1.1. One can also use 1.1 and the fact that any function in B(G) is a bounded pointwise l i m i t of a sequence of functions in Cc(G). Moreover, we may define the following notion of convergence in B(G,a) : a

62 sequence ( f n ) in B(G,~) converges to f c B(G,a) i f f

fn(X)

÷

f ( x ) f o r every x ~ G

and there e x i s t s h E B(G) such t h a t I f n l ~ h and I f l ~ h. Then fn + f " gn ÷ g ~ > f n gn ÷ f * g and f n* ÷ f * Let us define a representation of B(G,~) as a *-homomorphism L : B(G,a) ÷ ~ ( H ) , where H is a H i l b e r t space, which is continuous in the sense fn (L(f)E,n)

f o r any ~,n

÷

f

----->

~ H, and is such t h a t the line-at.span of { L ( f ) ~ ,

(L(fn

)~,~)

÷

f c B(G,~),

c H} is dense in H. 1.17. Lemma : Every representation of C (G,o) extends to a representation of B(G,o). C

Proof : Suppose t h a t f ~ B(G). There e x i s t s a sequence (fn) in Cc(G ) converging to f in B(G,o). By Lebesgue's dominated convergence theorem, f o r every ~,n in the space H of the representation L, f is i n t e g r a b l e w i t h respect to the measure ( L ( ) ~ , v ) (L(fn)~,~) ÷(L(f)~,n).

By the uniform boundedness theorem, L ( f )

and

is a bounded operator.

To show t h a t L is a *-homomorphism, we use again an approximation argument. The c o n t i n u i t y of L r e s u l t s from Lebesgue's dominated convergence theorem. Q.E.D The next goal is to r e a l i z e the inverse semi-group ~b of n@n-singular Borel G-sets (1.3.26) as an inverse semi-group of p a r t i a l

isometries.

For S ~ b

and

f ~ B(G), we define sf(x) = ~I/2(r(x),s)f(s-lx)~(s,s-Zx) = 0

if x # r-l(r(S))

if x ~ r-l(r(S)),

;

fs(x) = 61/2(d(x),s-l)f(xs-1)a(xs-1,s) = 0 s*f

if x ~ d-l(d(S))

= a(s-l,s)

if x e d-L(d(S)),

; and

(s-lf),

where ~ ( . , s ) denotes the v e r t i c a l

Radon-Nikodym d e r i v a t i v e of S. The notations have been

defined i n 1.1.11 and 1.1.18. For convenience, o ( s , s - l x ) o(sr(s-lx),s-lx). ~(s,t)

is w r i t t e n instead of

In accordance w i t h 1.1.18,

(u) = a ( u s , ( u . s ) t )

.

Also f o r a bounded Borel f u n c t i o n h on GO and f ~ B(G), h f ( x ) = hor(x) f ( x ) . t h a t s f , fs and s * f are functions in B(G).

We note

63

1.18. Lemma : The following relations hold in the *-algebra B(G,a) : (i) (ii) (iii) (iv) (v)

s(tf) = ~(s,t) (st)f for s,t C~b and f E B(G) ; fs~g = f.sg for f,g ~ B(G) and s ~ b (fs)*= s* f ' f o r

f ~ B(G) and s ~ b

; ;

s(f~g) = s f . g for f,g ~ B(G) and s e~b ; and fn ÷ f ---~Sfn ÷ sf for fn' f ~ B(G) and s ~ b -

Proof : The verifications are straightforward computations. (i)

s(tf) = ~I/2(r(x),s) tf(s-lx) a(s,s-lx) for x ~ r - l ( r ( s t ) ) = ~Z/2(r(x),s) ~i/2(r(x)-s,t) = al/2(r(x),st)

f(t-ls-lx)

f(t-ls-lx)

{(t,t-Zs-lx){(s,s-lx)

~(s,t)Qr(x) ~(st,(st)-l(x)

= ~(s,t) (st) f (x), and = 0 for x ~ r - l ( r ( s t ) ) . (ii)

fs.g(x) = ~fs(y) g(y-lx) ~(y,y-lx) d~r(X)(y) = ~ 61/2(d(y),s-Z)f(ys-l)g(y-Zx)

~(ys-l,s) ~(y,y-Zx)d~r(X)(y).

Changing the variable y into ys, this last expression becomes ~I/2(d(y).s,s-1)

~(d(y),s) f(y) g(s-Zy-Zx) ~(y,s)~(ys,s-ly-lx)

d~r(X)(y) = ~ 61/2(d(y),s) f(y) g(s-ly-Zx)~(y,y-lx)~(s,s-ly-lx)

d~r(X)(y)

= ~ f(Y) sg(y-lx) o(y,y-Zx) d~r(X)(y) =

(iii)

(fs)*(x)=

(x).

fs(x -1) ~-(x,x -I) =

al/2(r(x),s-l)

=

61/2(r(x),s -I) f-(x-ls -I) ~(x,x-ls -I) #(r(x)s-l,(r(x).s-1)s)

=

al/2(r(x),s-Z ) f (x-ls -1) ~(sx,x-ls-1)~(s-1 sx) ~(s-Z,s)or(x)

=

61/2(r(x),s -1)

=

(iv)

f .sg

~(s-l,s)

f~x-ls -I) m-(x-ls-I s) ~--(x,x-1)

f (sx){(s-l,sx) ~(s-l,s)or

(s-If)

s(f.g)(x) = ~l/2(r(x),s)

(x)

(x).

f.g

(s-lx)

=~61/2(r(x),s)

f(s-lxy) g(y-l) ~(xy,y-1) d~d(X)(y)

= jal/2(r(xy),s)

f(s-lxy) g(y-1) ~(xy,y-1) d~d(X)(y)

= sf,

g (x)

.

64 (v)

This is c l e a r , since we assume t h a t the v e r t i c a l

Radon-Nikodym d e r i v a t i v e

a ( . , s ) is bounded on compact sets. Q.E.D. 1.20. Lemma : Let L be a representation of B(G,~). (i)

There is a unique representation M of B(GO) such t h a t L ( h f ) = M ( h ) L ( f ) and

L(fh) = L(f)M(h) f o r every h ~ B(~ O) and every f ~ B(G). (ii)

There is a unique

G-representation V of the Borel ample semi-group ~b

as an inverse semi-group of p a r t i a l

isometries such t h a t L ( s f ) = V ( s ) L ( f )

and

L ( f s ) = L ( f ) V ( s ) f o r every s ~ ~b and every f ~ B(G). (iii)

The f o l l o w i n g covariance r e l a t i o n between V and M holds : V(s) M(h) V ( s ) *

= M(h s) f o r every s ~ ~b and every h e B(GO) where hS(u) = h(u s) i f u e r ( s ) , = 0 Proof : We f i r s t

note t h a t the approximate i d e n t i t y

if u # r(s).

constructed in 1.9, which can be

chosen countable since G is second countable, s a t i s f i e s

en . f

÷ f f o r any f ~ Cc(G,~ ),

Let L be a representation of B(G,~) on the H i l b e r t space H and l e t H0 be the l i n e a r span of { L ( f ) ~

: f e Cc(G), ~ E H}. We proceed as in 1.13 t o ' d e f i n e M(h) and V(s)

on H0 : n ( ~ m ( f i ) ~i ) = 1 n ( I k(fi)~i) =

M(h) V(s)

n ~ L ( h f i ) ~ i , and 1 n 1~ L ( s f i ) ~ i •

We check as in 1.13 t h a t M(h) and V(s) are well defined. (i) (ii)

This is obtained as 1.13. I t is immediate to check the f o l l o w i n g r e l a t i o n s on H0 :

V(s) V(t) = M ( ~ ( s , t ) ) V(st) V(s) - I

= M(-~s,s-l))

V(s -1)

V(s) V(s) -1 = M(Xr(S) ) and V(s) - I V(s) = M(×d(S) ), where ×A is the c h a r a c t e r i s t i c f u n c t i o n of A ; and

V(s)-1~ V(s)* In p a r t i c u l a r , (iii)

V(s) is a p a r t i a l

isometry and i t extends to H.

For s S ~ b , h E B(GO) and f ~ B(G)

65

s h-~-(s-l,s) s-1 f ( x ) = 6 1 / 2 ( r ( x ) , s ) = ~l/2(r(x),s) h(r(x).s) 7(s-l,s)

(h~(s-l,s))s-lf(s-lx)a(s,s-lx) i / 2 ( r ( x ) . s , s -1) f ( x ) ~ ( s - l , x ) a ( s , s - l x )

= h ( r ( x ) , s ) f ( x ) ~ ( s - l , s ) a(s,s -1) = (hSf) (x). Therefore, V(s) M(h) V(s)*

L(f) = V(s)r1(h) M(-~(s-ls))

V(s - I ) L(f)

= L(sh-~(s-l,s)s-lf) = L(hSf) = M(hs)

k(f). Q.E.D.

We have seen(1.7) that bounded representations of Cc(G,a ) could be obtained by integrating

a-representations of G. The correspondence between the unitary represen-

tations of a group and the representations of i t s

C *-algebra is well known and

j u s t i f i e s a large part of the theory of C*-algebras. The generalize~ion of this r e s u l t which we give in the case of groupoids has a more limited scope. We only consider groupoids which are second countable and representations on separable H i l b e r t spaces. Moreover, we need an additional assumption on the groupoid, namely, i t should admit s u f f i c i e n t l y many non-singular G-sets ( d e f i n i t i o n (1.3.27). This assumption is s a t i s f i e d in the case of a transformation group and of an r - d i s c r e t e groupoid. I do not have any example where i t is not s a t i s f i e d . The case of a transformation group is not new (e.g. [74], theorem 9.11, page 73). However, the proof usually given uses the standard Borel structure of the group and seems to f a i l in the case of a groupoid. Instead, we w i l l use part of a theorem of P.Hahn ([43], theorem 5.4, page 106), which w i l l be reproduced below as part of the proof of 1.21 since i t has not yet appeared in p r i n t . 1.21. Theorem :

Let G be a second countable l o c a l l y compact groupoid with Haar

system and with s u f f i c i e n t l y many non-singular Borel G-sets and a a continuous 2-cocycle in Z2(~,~). Then, every representation of Cc(G,¢ ) on a separable H i l b e r t space is the integrated form of a a-representation of Go

66 Proof :

We w i l l

only consider f a c t o r r e p r e s e n t a t i o n s . The general case is then o b t a i -

ned by d i r e c t i n t e g r a l decomposition and requires the d e f i n i t i o n of a d i r e c t i n t e g r a l of a f a m i l y o f

~ - r e p r e s e n t a t i o n s of G. Since t h i s theorem w i l l

only be used in the

case of f a c t o r r e p r e s e n t a t i o n s , we omit the general case here. Let L be a f a c t o r r e p r e s e n t a t i o n of Cc(G,o ) on the separable H i l b e r t space H. We use 1.17 to extend i t to a r e p r e s e n t a t i o n o f B(G,~) and 1.20 to o b t a i n the r e p r e s e n t a t i o n M o f B(~ O) and the ~ - r e p r e s e n t a t i o n V o f

~b such t h a t L(hf) = M(h)L(f) and L(sf) = V ( s ) L ( f ) .

r e s u l t s from m u l t i p l i c i t y

theory t h a t there e x i s t s a p r o b a b i l i t y measure ~ on GO and

a H i l b e r t bundle ~ o v e r

(GO,u) such t h a t M is u n i t a r i l y

It

e q u i v a l e n t to m u l t i p l i c a t i o n

on the H i l b e r t space F(~£) of s q u a r e - i n t e g r a b l e sections. From now on, ~e assume t h a t H = ~(~) and t h a t M is m u l t i p l i c a t i o n . a. Our f i r s t

task is to show t h a t the measure u is q u a s i - i n v a r i a n t . Let v be i t s

induced measure. We show t h a t f o r f E B(G), i f {u E GO

:

f = 0 v a . e . , then L ( f ) = O. Let E =

J l f l d ~ u > 0}. By assumption, M(XE) = O. I f x # r - 1 ( E ) ,

then f o r

eve~I g ~ B(G), f * g(x) = ~f(y) g(y-1) a ( y , y - l x ) d ~ r ( X ) ( y ) f,g

= 0 and t h e r e f o r e

= XE(f . g). Then

L ( f ) L ( g ) : L ( f , g) = M(×E) L ( f * g )

= O.

Since L is non degenerate, L ( f ) = O. Thus, f o r f ~ B(G), L ( f ) depends only on the v-class o f f . To show t h a t u is q u a s i - i n v a r i a n t , we pick a Borel set A in G of p o s i t i v e v-measure and we show t h a t i t has p o s i t i v e v-i-measure. We may assume t h a t f o r every . u e r(A) and any open set V in G such t h a t V n Au # ~, ~U(v m A) > N. Since G has sufficiently

many n o n - s i n g u l a r Borel G-sets, there e x i s t s a non-singular Borel

G-set S o f p o s i t i v e u-measure which is contained in A. We can construct a sequence (Un,en) where Un is a Borel set contained in A and en a non-negative f u n c t i o n in B(G) vanishing outside Un such t h a t fen d~ u = I f o r u e r(A) and every n and (Un) shrinks to S in the sense t h a t every neighborhood o f S contains Un f o r n s u f f i c i e n t l y Let f n ( y ) = ~ l / 2 ( r ( y ) , s )

en(Y ) f o r y E r - l ( r ( S ) ) ,

0 otherwise. Then, f o r every

f ~ Cc(G), fn

~ f(x) =J~m/m(r(y),s) =J~l/2(r(x),s)

en(y ) f ( y - l x ) en(Y ) f ( y - l x )

~ ( y , y - I x)d~ r(x) (y) ~(y,y-lx)d~r(X)(y).

large.

67 Hence, f o r every x, fn

*' f ( x ) ÷ ~ i / 2 ( r ( x ) , s )

Moreover Jfn ~" f l ( x )

f(s-lx)

_< J s f I ( x ) .

in the weak operator topology.

a(s,s-Zx) = s f ( x )

Therefore, L(fnO)L(f) = L ( f n . f ) ÷ L ( s f )

= V(s)L(f)

I f A had zero v - I - measure, then since supp f n c f ~ l ,

.¢-x-

we would have 'n = 0 v a.e. and L ( f n ) * = L ( f n ) = O, hence L ( f n ) = O. We would conclude t h a t V ( s ) L ( f ) = 0 f o r every f ¢ Cc(G ), hence V(s) = O. However t h i s would c o n t r a d i c t the f a c t t h a t V(s)V(s) = M(×r(S) ) > O. b.

Let us show next t h a t f o r each n o n - s i n g u l a r Borel G-set S, the p a r t i a l

V(s) on

isometry

r(~) is of the form

V(s)~ (u) = z~l/2(us,s) c ( u , s ) ~(u-s) = 0 where & ( ' , s )

f o r u ~ r(S) f o r u f r(S)

is the h o r i z o n t a l Radon-Nikodym d e r i v a t i v e of S (1.3.18) and c ( u , s ) ,

defined f o r ~ a.e. u E r(S) is an isometry of ~Z~u.s onto~Zuu. This f o l l o w s d i r e c t l y from a r e s u l t of Guichardet ( [ 3 8 ] , Let~and

p r o p o s i t i o n i , page 82) which we r e c a l l

here :

~ b e two H i l b e r t bundles over the standard measure spaces (X,m) and

(Y,B) r e s p e c t i v e l y ,

¢ an isomorphism of (X,m) onto (Y,6) and U an isometry of r(}~)

onto r(~) s a t i s f y i n g UM(h) U-1 = M(ho¢ -1) f o r every h ~ L~(X,~), where M denotes the m u l t i p l i c a t i o n ~Z~_l(y)Onto ~y defined f o r B a . e , U~ (y) = r l / 2 ( y )

operator. Then, there e x i s t isometries u(y) from y such t h a t f o r every ~ ~ r ( ~ ) ,

u(y) ~ ( ¢ - 1 ( y ) ) B a . e .

where r = dCm is the Radon-Nikodym d e r i v a t i v e of ¢~ w i t h respect to B. dB c.

Next, we show t h a t the set of constant m u l t i p l i c i t y

f o r p = 1,2 . . . . . ~, of the H i l b e r t b u n d l e ~ i s A were not almost i n v a r i a n t , t h a t f o r every x E B,

almost i n v a r i a n t ( d e f i n i t i o n

1.3.5).

If

there would be a Borel set B of p o s i t i v e ~-measure such

r ( x ) c A and d(x) # A. By assumption, B contains a n o n - s i n g u l a r

Borel G-set S such t h a t ~ ( r ( S ) } > O. However f o r c ( u , s ) from ~Ju.s onto ~ u '

d.

A = {u e GO : dim~Q = p}

~J a.e. u E r ( S ) , there is an isometry

hence u-s E A. This is a c o n t r a d i c t i o n .

We show t h a t f o r a Borel set B in GO, the p r o j e c t i o n M(XB) is in the commutant

68

of L i f f

B is almost i n v a r i a n t . Suppose t h a t there e x i s t s a Borel subset A of G such

t h a t M(B) L(XA) # L(×A) M(B). Then

v(A n r - l ( B )

& A n d - l ( B ) ) > 0 : B is not almost

i n v a r i a n t . Conversely, i f B is not almost i n v a r i a n t , then e i t h e r r - 1 ( B ) \ d - l ( B ) d'1(B)\r-l(B)

has p o s i t i v e

or

v-measure and contains a non-singular Borel G-set S with

~ ( r ( S ) ) > O. Then M(×B)V(s ) = V(s) and V(s)M(×B) = O. Since V(s) is the weak closure of {L(f)

: f ~ Cc(G)} , there e x i s t s f c Cc(G) such t h a t M(XB)L(f ) ~ L(f)N(×B). Since

we assume t h a t L is a f a c t o r r e p r e s e n t a t i o n , t h i s shows t h a t ~ is ergodic. From c, we conclude t h a t the H i l b e r t bundle~Y~is homogeneous, hence isomorphic to a constant H i l b e r t bundle, so t h a t we can w r i t e H = L2(GO,u,K). e.

We show t h a t L s a t i s f i e s the i n e q u a l i t y

l(L(f)~,~)I

~ jlfldv 0

II~!l l!nll

the constant section u measure f ÷ ( L ( f ) ~ , n )

f o r every ~,n ~ K where ~ ~ K is i d e n t i f i e d w i t h

÷ ~. Let ~ and n be f i x e d u n i t vectors in K. Since by a. the is a b s o l u t e l y continuous with respect to VO, there e x i s t s a

Borel f u n c t i o n c such t h a t ( L ( f ) ~ , n ) to prove t h a t I c ( x ) l < I Ic(x)l

= if(x)

c(x)dvo(x ) f o r every f ~ B(~). We have

v a.e. I f not, there e x i s t a > I such t h a t

> a} > 0 and we may f i n d a Borel set A contained in {x c G

v{x e G :

:Ic(x)l

> a}of

p o s i t i v e v-measure and such t h a t f o r u E r(A) and every open set V which meets p u, ~U(v n A) > O. Proceeding as in a, we f i n d a n o n - s i n g u l a r Borel G-set S o f p o s i t i v e u-measure contained in A and a sequence (Un,en) where Un is a Borel set contained in A and e n a non-negative f u n c t i o n in B(G) vanishing o f f Un such t h a t (i) (ii) (iii)

~e dXu = I f o r u ~ r(A) n

"

Un shrinks to S when n ÷ ~ ; and f o r every y c Un, D ( s ' l y ) ~ b 2 where i < b < a.

I t is possible to f u l f i l l

t h i s l a s t c o n d i t i o n because any neighborhood of a subset

o f GO of p o s i t i v e ~-measure contains a subset of p o s i t i v e v-measure where D ~ b2. Let f n ( y ) = ~ I / 2 ( r ( y ) , s )

en(Y)I~(y)/Ic(y)l.

Then

( L ( f n ) ~ , n ) = J f n ( y ) c(y) D-m/m(y)dv(y) = ~61/2(r(y),s) =#51/2(u,s) =~f&l/2(u,s)

D-m/2(r(y)s) e n ( Y ) I c ( y ) I D - 1 / 2 ( s ' l y )

D-1/2(us)

en(y)Ic(y)I

en(Y)Ic(y)I

D- I / 2

dr(y)

D-1/2 ( s - l y ) d~U(y) du(u)

( s - l y ) d~U(y) d~(u)

69 by (1.3.20) and this dominates ab-1

/

A1/2(u,s) du(u).

r(S) On the other hand, we know from b that V(s)~(u) = A1/2(u,s) c(u,s) ~(u's) for u~ r(S) with c(u,s

isometry of~{o, s into ~u" So

(V(s)~,~) = J ~ l / 2 ( u , s ) ( c ( u , s ) ~ , n )

d~(u) ; and

I(V(s)~'n)I # ~r(S) &l/2(u's) du(u). This is a contradiction because L(fn) : M(hn) V(s), where

hn(u ) =fen(Y) E(}]/ Ic(y)I d~U(y) satisfies

]hnl ~ ~ I,

tends to zero in the weak operator topology. Indeed, f~ . f ( x ) - h~sf(x) =/e ( y ) i ~ i

~l/2(r(x),s)[o(y,y-lx)f(y-lx)

~(s,s-lx)f(s'Ix~

-

dxr(X)(y)

tends to zero for every x e G and every f e Cc(G), and If a * f ( x ) l f.

~ l(sf)(x)I

and lhmsf(x)l ~ ] ( s f ) ( x ) I .

The conclusion is given by the following lemma, due to P.Hahn ( ~ 9 ] ,

theorem

5.4, page 106). Lemma (P. Hahn) : Let G and ~ be as above. Let L be a representation of Cc(G,~) on L2(GO,~,K) where u is a quasi-invariant probability measure and K a separable Hilberi space, such that (i)

l(L(f)~,n)I

~JIfldv 0

II~II

Ilqll

for every

E,n~K

( ~ also denotes the constant function ~ L 2 ( G O , u , K ) ) . (ii)

L(hf) = M(h)L(f) for every h ~ Cc(GO) and every f e Cc(G), where M is

multiplication. Then, L is the integrated form of a ~-representation of G on the constant Hilbert bundle with f i b e r K over (GO,u). Proof : a.

There exists a weakly measurable map x ÷ A(x) of G into the bounded operators

70 on K of norm < i such that (L(f)~,~)

= Jf(x)

(A(x)~,n)

du0(x ) for every ~,n c K.

For the condition ( i ) means that f ~

(L(f)~,n)

is a bounded linear functional

LI(G,~o) of norm ~ II~II llnII. This gives a map (~,n) This map is sesquilinear and s a t i s f i e s into the space ~ ( G , v )

~

on

k(~,n) of K × m into L~°(G,~).

Ik(~,n)l~ ~ ll~II IInll. Using a l i f t i n g

of L~(G,v)

of bounded measurable functions, we obtain for each x c G

a bounded sesquilinear functional

on K × K of norm ~ 1, hence an operator A(x) of norm

< 1. The map x ~ A(x) has the required properties. b.

For every ~,n ~ L2(~O,~,K) and for every f E Cc(G ), (L(f)~,n)

Yf(x)

=

(A(x)~ od(x), nor(x)) duo(X).

Since both sides are bounded sesquilinear functionals

on L2(G 0 ,~, K) (cf. 1.7) i t

suffices to check the equality on the algebraic tensor product L2(G0,~) ® K and by sesquilinearity

on elements of the form h(u)~, where h E Cc(G0 ) and ~ ~ K :

(L(f)h~,kn)

= (L(f)M(h)~,

M(k)n)

= (k(k* fh)~,n) by ( i i )

:jk-

or(x) f ( x ) hod(x) (A(x)~,~) dvo(X )

=J f ( x ) c.

(A(x) hod(x)~, kor(x)n) du0(x )

A s a t i s f i e s A ( x ) * = # ( x , x -1) A(x -1) for u a . e . x . (k(f*)~,n)

= J f*(x)

For all (,n e K and f s Cc(G )

(A(x)~,n) dvo(X )

= ~ f ( x - I ) T(x,x -1) (A(x-Z)(~,n) dvo(X ) = ~ f ( x ) @(x,x-1)(A(x'l)(E,n)

(~,L(f)m) Hence ( A ( x ) * ~ , n )

= f f ( x ) (~, A(x)n)

dvo(X ) (because VO is symmetric), and

dvo(X )

= ~(x,x -1) (A(x)~,n) for u a.e.x.

Since K is separable, we obtain

the result. d.

The function A s a t i s f i e s A(x)A(y) = ~(x,y) A(xy)

For all ~,n ~ L2(GO,~,K) and f,g L(g)~(u) = j g(y) (k(f)L(g)~,n)

~ Cc(G )

A(y) ~od(y)

= Jf(x)g(y)

u 2 aoe. ( x , y ) .

D-I/2(y)

d~U(y) for

(A(x)A(y)~od(y),nor(x))

a.e. u by b and D'I/2(xy)

D-l(y) du2(xy).

(These computations have already been done in 1.7.) On the other hand,

71

(L(f * g)g,q) = ~ f ( x y ) g ( y ' 1 ) o ( x y , y - I ) A f t e r the change of variable f ( x ) g(y) ~(xy,y -1)

(A(x)~ o d ( x ) , nor(x)) D1/2(x) dv2(x,y).

(x,y) ÷ ( x y , y - 1 ) , (A(xy)~od(y),

this is equal to

qor(x)) D1/2(xy) D-I(y) d~2(x,Y)

Using the density of Cc(G ) ® Cc(G ) in Cc(G2), we obtain f o r all ~,q e K (A(x)A(y)~,q)

= ~(x,y)

(A(xy)~,q)

for

v 2 a.e. (x,y)

since K is separable, we obtain our r e s u l t . e.

Let S be a non-singular Borel G-set. Since A cannot be evaluated on S, we consider

instead a function B defined as follows. a(xy,y " I ) A(x) for

I t results from d that A(xy)A(y -1) =

v 2 a.e. (x,y) or e q u i v a l e n t l y ,

Let B(x) = f A(xy) A ( y - l ) ~ ( x y , y -1) f ( d ( x ) , y )

for v a.e. x and ~d(x) a . e . y .

d~d(X)(y), where f is a p o s i t i v e measu-

rable function on GO x G such that Jf(u,y)

d~U(y) = 1 for every u E GO.

Then, B(x) = A(x) f o r for v a.e.x.

v a.e.x.

Let us show that B(xs-1)B(s) = a ( x s - l , s )

(As usual s in B(s) and in

o(xs-l,s)

stands f o r d ( x s - l ) s ) .

B(x) By quasi-

invariance of u under S, i t results from d that for v a.e. x c d - l ( d ( s ) ) A(xs-l)A(y)

= ~(xs-l,y)A(xs-ly)

f o r xd ( x s - l )

B(xs -1)

= A(xs-1). Therefore

a.e. y and

B(xs-1)B(s) = A(xs -1) J A(sy)A(Y - I ) E ( s y , y - 1 ) f ( d ( x ) , y ) d ~ d ( X ) ( Y ) = A(xs " I ) f A(y)A(Y - I s ) =]A(xs'ly)A(y-ls)

~(y,y-Zs)f(d(x),s-Zy)

d~d(x)'s-Z(y)

o(xs-l,y) ~-(y,y-ls)f(d(x),s-ly)

=SA(xy)A(y-1)o(xs-I,sy)

~(sy,y-I) f(d(x)'y)

d~d(x)'s-l(y)

d~d(X)(Y)

= o(xs'Z,s)B(x). f.

A(x) is a unitary operator f o r u a . e . x .

selection lemma. We w i l l

Hahn's proof uses the von Neumann

use instead the existence of s u f f i c i e n t l y

G-sets. The set E = {x E G : B(x) is not unitary} i t has positive

is a measurable set. Suppose that

v-measure. Then, i t contains a non-singular Borel G-set S of p o s i t i v e

-measure. Let us define v l ( s ) on L2(GO,~,K) by (,)

vl(s)

many non-singular

(u) = AZ/2(u,s)B(us)~(u's) = 0

i f u ~ r(S) and otherwise.

72 Then for every f s Cc(G), L(f)vl(s)~(u) for ~ a,e.

= J f ( x ) B ( x ) A l / 2 ( d ( x ) , s ) B(dx)s) ~(d(x).s)D-1/2(x ) d~U(x)

u.

We change the variable x into xs -1 to obtain ~f(xs-1)B(xs - I ) A l / 2 ( d ( x ) . s - l , s ) B ( s ) ~ ( d ( x ) )

D-1/2(xs - I ) 6(d(x),s-1)d~U(x).

We use A(u,s-1)D(us - I ) = a(u,s - I ) for u a.e. u (1.3.20) to obtain 5 f ( x s -1) a l / 2 ( d ( x ) , s - Z )

B(xs -1) B(s) ~(d(x)) D-i/2(x) d~U(x).

Finally by d, this yields J f(xs - I ) 61/2(d(x),s -1) a ( x s - l , s ) B(x) = I f s ( x ) A(x)

~od(x) D-I/2(x) d~U(x)

~od(x) D-I/2(x) dxU(x)

= L(fs)~(u). Hence L ( f ) v l ( s )

= L(fs) = L(f)V(s) for every f, so that VI(s) = V(s).In particular

vl(s) is a non-zero partial

isometry with range M(×r(S) ). Comparing (*) and b of the

proof of the theorem, we see that B(us) is a unitary operator for This is a contradiction.

Hence, for

~ a.e. u in r(S).

~ a.e. x, B(x) and consequlntly A(x) are unitary.

I t suffices to modify A on a null set so that i t becomes unitary for every x to fulfill

all the conditions of d e f i n i t i o n 1.6. Q.E.D.

1.22. Corollary : Under the assumptions of the theorem, every representation of Cc(G,a ) on a separable Hilbert space is bounded. 1.23. Corollary : Under the assumption of the theorem, the integrating process 1.7 establishes a b i j e c t i v e correspondence between (a,G)- H i l b e r t bundles and separable Hermitian C*(G,a)-modules which preserves intertwining operators. Proof : We have to prove that two a-representations (~,J£~L) and ( u ' , J ~ , L ' ) which give u n i t a r i l y equivalent integrated representations are equivalent. Let L and L' be representations of Cc(G,a ) on

r(J~) and F ( ~ ) whose r e s t r i c t i o n s

tions M and M'. I f # is an isometry of

F(~C) onto r ( ~ )

to Cc(GO) are multiplica-

which intertwines L and L ' ,

i t also intertwines M and M'. Therefore, the scalar spectral measures u and and M' are equivalent and there exists a measurable f i e l d u ~

u' of M

#(u) where #(u) is an

73 isometry of ]~u o n t o ~ ,

decomposing

~. The relation

{L(f) = L'(f)~ becomes

~f(x)(~ or(x)L(x)~od(x), nor(x))d~o(X ) = ~ f ( x ) ( L ' ( x ) ~ d ( x ) ~ o d ( x ) , n o r ( x ) ) we have assumed ~ = ~'. This gives ¢or(x) L(x) = L'(x){od(x)

d~o(X ) where

for v a . e . x . Q.E.D.

The o-representations of a group G can be considered as ordinary representations of the extension group G°.

This leads to an alternate definition of C*(G,o), for a

groupoid G. Let Ga denote the extension]~x G of -[by G defined by the 2-cocycle o ~ Z2(G,q[). Recall (1.1.12)that (s,x) ( t , y ) = (sto(x,y),xy) ( s , x ) - I = ( s - l o ( x , x - 1 ) - l , x -1) Its unit space can be identified with GO. I t is a locally compact grou:ooid with the product topology and i t has the l e f t Haar system {h x ~u}

where h is the Haar

measure of 7[. 1.22. Proposition : I

The C*-algebra C*(G,o) is the quotient of C*(G°) by the kernel

of the representation L of C*(G°)

obtained by integration

which satisfy

L(tf) = tL(f) for any t sT, f ~ Cc(G° ) and where t f ( s , x ) = f ( t ' l s , x ) . Proof :

The map ~ from Cc(G°) to Cc(G,o ) given by the formula ~f(x) = ] f ( s , x )

is a *-homomorphism. Indeed, ~(f*g)(x) = f f * g ( s , x ) s d s =J]J f ( s t o ( x , y ) , x y ) g ( t -i"o(y,y -1 )-Z,y-l)dtd~d(X~(y)sds. One makes the changes of variable u = sto(x,y) and v = t - l a ( y , y - l ) -I to obtain jJf

f(u,xy) g(v,y -1) uvo(y,y-1)o(x,y)-Zdudvdxd(X)(y)

=J(jf(u,xy)udu)

( j g(v,y-1)vdv)o(xy,y -1) d~d(X)(y)

= ~(f) . ~(g) (x). Moreover, ~(f*)(x)

=#f*(s,x)sds = ] - f ( s - 1 a ( x , x - l ) -1,x-1)sds =~ f ( t , x -1) t dt ~(x,x -1) = ~(f)(x - I ) o(x,x -1)

sds

74 = ~T(f) Since

(X) ,

~ is bounded with respect to the L I norms, i . e .

ll~(f)IIl ~

!IfIll, because

~JJ

f l~(f)(x)[d~U(x)

[f(x,s) I

dsd~U(x), i t follows that i t is bounded

with respect to the C*-norms and extends to a *-homomorphism from C * ( G ° ) to C*(8,~).

I t is onto since i t s image is closed and contains Cc(G,o ). I f L is a repre-

sentation of C~ (G,~), Lo~ Lo~ ( t f ) = k ( t ~ ( f ) )

is a representation of C* (G~) which s a t i s f i e s

= t

Lo~(f)

since

~(tf) =

t~(f).

Conversely, i f L is a representation of C* (G°), s a t i s f y i n g t h i s r e l a t i o n and which is of the form (*)

(L(f)E,q) = J J f ( s , x )

(L(s,x)~od(x), nor(x)) dv0(x)ds ,

then L(s,x) s a t i s f i e s L(ts,x) = tL(s,x) f o r h ×h a.e. ( t , s ) and ~ a.e. x, as one sees from the equation L(gf) = ~(g)L(f) where g e C c ( T ) , (gf)(s,x) = ~g(t)f(t-ls,x)dt

and ~(g) = ~ g ( t ) t d t .

L(s,x) can be replaced by L'(s,x)

:JL(ts,x)tdt

without changing ( , ) . Then i t s a t i s f i e s L(ts,x) = tL(s,x) (L(f)~,n)

=jj

f o r every ( t , s ) and v

a.e.x.

f(s,x)sds (L(e,x)~od(x),nor(x))

so that L factors through

Thus

d~0(x),

~. Q.E.D,

2.

Induced Representations

Let G be a l o c a l l y compact groupoid with Haar system {~u} and H a closed subgroupoid G containing GO and admitting a Haar system { ~ } , 2-cocycle. Just as in the case of groups, a

and ~ ~ Z2(G,T) a continuous

u-representation of H may be induced to a

75 o - r e p r e s e n t a t i o n of G, We d e s c r i b e t h i s process below, R i e f f e l ' s ced r e p r e s e n t a t i o n s is p a r t i c u l a r l y

version [54] o f indu-

w e l l adapted to t h i s c o n t e x t and t h i s s e c t i o n

i m i t a t e s the e x p o s i t i o n he gives in the case o f groups. We f i r s t

need some t o p o l o g i c a l

r e s u l t s which are well known i n the group s i t u a t i o n . 2.1. P r o p o s i t i o n : x ~ y iff (i) (ii) (iii)

Let G and H be as above and consider the r e l a t i o n on G defined by

d ( x ) = d(y) and xy -1 ~ H. I t is an equivalence r e l a t i o n . The q u o t i e n t t o p o l o g y on the q u o t i e n t space H\G is Hausdorff. The q u o t i e n t map r : G ÷ H'G is open.

( i v ) The q u o t i e n t space H\G is l o c a l l y

compact.

(v) The domain map d induces a continuous and open map from H\G onto GO. Proof : (i) (ii)

Clear. Since H i s c l o s e d , the s e t { ( x , y )

G x G. The graph o f the r e l a t i o n homeomorphism ( x , y ) , + (iii)

(x,y -I)

~ G2 : xy ~ H} is closed i n G2, hence in

i s the image by e o f t h i s s e t , where e i s the

: G ×G÷ G x G

Let 0 be an open s e t G ; we have to show t h a t i t s s a t u r a t i o n HO is also open.

Let hx be a p o i n t in HO w i t h h ~ H and x c O. There e x i s t s a noncnegative f u n c t i o n ¢

E Cc(H ) such t h a t

¢(h) ~ 0 and a non-negative f u n c t i o n g E Cc(G ) such t h a t

g(x) # 0 and supog c O. The same argument as in 1.1 shows t h a t the f u n c t i o n

¢-g

defined by ¢'g (Y) = I ¢ ( k )

g(k-ly)

dX~ (y)

(k)

i s continuous on G ; t h e r e f o r e {y : ¢ . g ( y ) # O} is an open set ; since i t contains hx and is contained in HO, we are done. ( i v ) This r e s u l t s from ( i i )

and ( i i i ) .

(v) This is c l e a r since d : G + GO i s continuous and open. Q.E.D. 2.2. Lemma :

There e x i s t s a Bruhat approximate c r o s s - s e c t i o n f o r G over H\G, t h a t

i s , a non-negative continuous f u n c t i o n b on G whose support has compact i n t e r s e c t i o n w i t h the s a t u r a t i o n HK o f any compact subset K of G and is such t h a t f o r every x e G,

I

b(h-lx)

dz~(X)(h)= M

I.

76 Proof :

By [ ~ ,

Lemme 1, page 96, there e x i s t s a non-negative continuous f u n c t i o n

g

non-zero on every equivalence class and whose support has compact i n t e r s e c t i o n w i t h the s a t u r a t i o n of any compact subset of G. The g ( h - l x ) d ~ (x) (h) is continuous and s t r i c t l y

f u n c t i o n gO defined by g°(x) = positive.

The f u n c t i o n b = g/gO

is a Bruhat approximate cross section f o r G over H\G. Q.E.D. 2.3. Proposition : Let G and H be as above and consider the r e l a t i o n defined by ( x , y ) ~ ( x ' , y ' ) (i)

iff

y = y' and xx ' - I

~ on G2

E H.

I t is an open Hausdorff equivalence r e l a t i o n and the q u o t i e n t space H\G2

w i t h q u o t i e n t topology is l o c a l l y compact. (ii)

The r e l a t i o n

is compatible

with the groupoid s t r u c t u r e of G2, so

that

HSG2 is a l o c a l l y compact groupoid, i t s u n i t space may be i d e n t i f i e d w i t h HIG. (iii) Proof

The groupoid H\G 2 has a Haar system, namely

{6~ × ~d(~), ~ ~ H\G}.

:

( i ) This is v e r i f i e d as 2.1, in f a c t H\G2 = { ( ~ , y ) E H\G x G : d(~) = r ( y ) } . (ii)

The composable pairs in G2 are ( ( x , y ) , ( x y , z ) ) .

then ( x ' y , z ) ~ ( x y , z ) and ( x ' , y )

(x'y,z)

Therefore i f

(x',y)

= (x',yz) ~ (x,yz) = (x,y)(xy,z).

~ (x,y), Hence we

may define the f o l l o w i n g groupoid s t r u c t u r e on H\G 2. The composable pairs are ( x , y ) , (~,z),

(~,y) (x-y,z) = (½,yz) and the inverse of ( x , y ) is ( x ~ 1 , y - 1 ) . F i n a l l y , by

definition

of the q u o t i e n t topology, the m u l t i p l i c a t i o n

and inverse maps are continuous.

Since (~,y) ( # , y ) - 1 = (#,yy-1) = ( x , d ( ~ ) ) , we may i d e n t i f y

the u n i t s~.ace of H\G2

and H\ G. (iii)

This is c l e a r ; here J f

~

×d~ d(#) = I f ( ~ , y ) d x d ( X ) ( y ) Q.E.D.

Notation : Let o be a continuous 2-cocycle in Z2(G,-~), one can associate w i t h i t continuous 2-cocycles on H and on H\G2 r e s p e c t i v e l y in the f o l l o w i n g way. On H, denotes i t s r e s t r i c t i o n

to H. On H\G 2, ~ is defined by ~ ( x , y , z ) = ~ ( y , z ) (we w r i t e

( x , y , z ) instead of ( ( x , y ) ,

(~,z))).

The cocycle property is e a s i l y checked.

For# ~ Cc(H,a ) and f c Cc(G), l e t us define

77 • f (x) = J ¢(h) f ( h - l x )

~(h,h-lx) dx~(X)(h), and

f • ¢ (x) = # f(xh) ¢(h -I) ~(xh,h -I) dz~(X)(h . For ~ m Cc(HIG2,o ) and f c Cc(G), let us define • f (x) = ] ¢ ( x - l , x y )

f(y-l)

o(xy,y-l)dxd(X)(y),

f • ~ (x) = I f(Y) ~ ( Y , y - l x ) ~ ( Y , y - I x ) d~r(X)(y)

and

.

2.4. Proposition : ( i ) The space Cc(G ) is a Cd(H,~)-bimodule and a Cc(H~G2,o)-bimodule ; and the actions of Cc(H,{ ) and Cc(H~,G2,~) on opposite side commute. (ii)

The algebra Cc(H,~ ) acts as a *-algebra of double centralizers

algebra Cc(G,~), this action extends to the C*-algebra *-homomorphism of C* (H,~) into the m u l t i p l i e r

on the

C*(G,~) and gives a

algebra of C (G,~).

Proof : (i) One has f i r s t

to check that, with above notations, Cf, f¢, m-f and f.#

are

indeed in Cc(G ). This is done in exactly the same fashion as in proposition 1.1. The verification (¢ * 9)

of the various a s s o c i a t i v i t y -f=¢-(~

relations,

.f) for f c Cc(G )

and ¢,# both in Cc(H,{ ) or in Cc(H\G2,~), the analogous r e l a t i o n for the action on the r i g h t , and • (f • ~) = (¢ • f) • ~ for f ~ Cc(G ) and ¢,~

in C (H,~) or in C (H\G2,~), is straightforward

but tedious. Let us check

one of them as an example. Suppose f ~ Cc(G ) and ¢,~ c Cc(HXG2,~). Then • ~ (x,y) = f ¢(x,yz)~(xyz,z -1) o(yz,z-m)dxd(Y)(z), f .(¢.~)

(x) = Jf(y)¢ =#f(y)

and

~(~,y-lx) ~(y,y-lx)dlr(X)(y) ¢(!},y-mxz) ~(~z,z -1) ~ ( y , y - l x )

~(y-mxz,z-m)

• d~d(x) (z)d~r(x) (y) =#f(y)

#(y,y-lxz)

~(x~,z - I ) ~(y,y-mxz) ~(xz,z -1)

d~r(x) (y) d~d(X)(z) =If

• ¢(xz) ~ ( ~ , z - I ) ~(xz,z -1) d~d(X)(z)

=J f • ¢(z) # ( z , z - l x )

d(z,z-lx)dxr(X)(z)

78 = ( f • #) • ~ (x) . (if)

We have to check the equations

f~(¢.g)

= f • ¢ * g and ( # . f)~'

= f*.

#~

f o r f , g e Cc(G) and ¢ E Cc(H). This is done as above. To prove t h a t t h i s action extends extends to C * ( G , { ) ,

one can introduce the Banach algebra L i , r ( G , a ) , the completion

of Cc(G,a ) f o r the norm ll l[i, r. I t has a bounded l e f t approximate i d e n t i t y . Thus, i f L is a bounded representation of Cc(G,a ), there is a unique bounded representation LH, called the r e s t r i c t i o n

of L to C*(H,~), such t h a t L ( ¢ - f ) = LH(#)L(f ) and

L ( f . # ) = L ( f ) LH(#). What makes the proof go is the i n e q u a l i t y I I # . f l I i , r ~ I I # j I i , r I I f I l l , r which is obtained as in 1.6. This gives a f a i t h f u l ~-homomorphism of Cc(H,a) into the m u l t i p l i e r algebra of C * ( G , a ) which is norm-decreasing when Cc(H,c ) has thell III norm. Hence i t extends to a *-homomorphism of C*(H,~) i n t o the m u l t i p l i e r algebra of

C * (G,~). Q.E.D. Let X = Cc(G), B = Cc(H,~ ) and E = Cc(H\G2,~) ; view X as a l e f t E- and r i g h t B-bimodule. One would l i k e to e x h i b i t X as an E-B i m p r i m i t i v i t y bimodule ( d e f i n i t i o n 6.10 of

[64).

I did not succeed in doing that except in p a r t i c u l a r cases. The

candidates f o r E and B-valued inner products on X are

\

B (h) = j ~(x -1) g(x-Zh)~(x,x -1) ~ ( x , x - l h ) d x r ( h ) ( x ) E(X,x-ly)

= ~ f(x-lh) g(y,h) ~(y-lh) ~(y'lh,h-ly)

and

a(x-lh,h-ly)dx~(X)(h).

(By l e f t invariance of the Haar system, the r i g h t hand side depends on x only). The algebraic r e l a t i o n s
, gb> B =
~g>Bb

<ef,g> E

= eE

E

= < f b * 'g>E

<ef'g>B

= < f ' e ~ g>B

fB

= E f '

(f,g

c X, b ~ B) (e E E)

( f ' g' f ' c X)

are s a t i s f i e d , as one may check in the same fashion as above. 2.5. Lemma : ( i ) The l i n e a r span of the range of <'>E contains an approximate l e f t i d e n t i t y

79 f o r Cc(HIG2, ~) with the inductive l i m i t topology. (ii)

A s i m i l a r statement holds f o r <'>B and Cc(H,e ).

Proof :

I t is the same proof as in proposition 7.11 of

~3]

(and in lemma 2, page

201, of ~ 9 ] ) . ( i ) Let C be a compact subset of H\G and c a p o s i t i v e number.Choose a compact set K in G such that ~(K) = C. There e x i s t s a d - r e l a t i v e l y borhood N of GO in G such that l o ( x - l , y )

- 11

compact (see 1.9) neigh-

< ~ f o r y ~ ~, x ~ K and r ( y ) = r ( x ) ,

because ~ is continuous and takes the value I whenever one of i t s arguments is a u n i t . There is a l o c a l l y f i n i t e such that v i l v i

cover of G consisting of open r e l a t i v e l y compact sets (Vi)

c N and a p a r t i t i o n of unity subordinate to i t .

Multiply this

p a r t i t i o n pointwise with a Bruhat approximate cross-section b which has been truncated so that b ~ Cc(G ) and f b ( h - l x ) d ~ ( X ) ( h )

= 1 f o r x c K, and obtain a f i n i t e

number of non-negative functions of f l . . . . . fn c Cc(G) such that suppf i ~ Vi and n f fi(h'lx)d~(X)(h) = 1 f o r x c K. For each i , choose gi ~ Cc(G) such that i=l suppg i ~ Vi , ~ I g i ( Y ) I d ~ U ( y ) = 1 f o r u ~ r {x : f i ( x ) ~ 0}, and [ g i ( y ) I = ~i(y ) ~(y-ly). Let e fc c NI = ~ < ~ , g i > , where f ( x ) = f ( x ' l ) . ~~ ~''J i=1 < f i ' g i >(~'y) = ] f i ( h - I x ) gT (h-mxy) ~ ( y - l x - l h ' h - m x y ) { ( x - l h ' h - l x y ) d ~

Since ( x ) ( h ) ' e(c,e,N)

satisfies (a) 0 (c,~,N)( x 'YJ~ = 0 i f y # N, and (b) l]O(c,~,N)(~,y)d~d(X)(y)

-11 L ~ i f x cC.

I t r e s u l t s from the proof of proposition 1.9 that the net { e ( c , ~ , N ) ) d i r e c t e d by ~,~,N)<(c',~',N')

iff

c ~c',

c >c' and N ~ N '

is a l e f t approximate i d e n t i t y f o r

Cc(H\G2,o). (ii)

This is done in a s i m i l a r manner. Let K be a compact subset of GO, e a

p o s i t i v e number and N an r - r e l a t i v e l y an r - r e l a t i v e l y

compact neighborhood of GO in G, One can f i n d

compact neighborhood U of GO and non-negative continuous functions f

and g on 6 such that UU- 1 ~ N, the support of g is compact and contained in U, while g(x)d~U(x) = 1 f o r u ~ K, the support of f is contained in U and has compact i n t e r s e c t i o n with the saturation HL of any compact subset L of G, while If(h-lx)

dl~(X)(h) = 1 f o r x ~ r - l ( m ) n U, and I # ( h - m x , x - l h ) o ( h - l x , x -1) - 1! ~

when x ~ U n r - l ( K ) ,

h - l x ~ U and h ~ H. Then, one notes t h a t the function @(c,~,N)'

80 defined by <~(N,E,K)(h) = B(h) =If(hx)

g(x) { ( h x , x - l h -1) a ( h x , x - 1 ) d ~ d ( h ) ( x ) ,

satisfies (a)

¢(N,E,K)(h) = 0 i f h ~ N and

(b) [/¢(N,c,K)(h)dXHu(h ) - 1] E c i f u E K. Therefore, the net {~(n,E,K)} is a r i g h t approximate i d e n t i t y

f o r Cc(H,d ). Q.E.D.

2.6. C o r o l l a r y

:

( i ) The l i n e a r span of the range of <'>E is dense in Cc(H\G2,~) and in C* (H\G2,a) The l i n e a r span of the range of <'>B is dense in Cc(H,o ) and in C ~ (H,o).

(ii) Remark

:

I t seems d i f f i c u l t

in lemma 2, page 201, of [39].

to construct approximate i d e n t i t i e s

as those obtained

In the general case, with the notations of the proof,

one would need sets V.'sl such t h a t r ( V i ) = GO. In the case when H = G0, i t is not hard to carry the proof through. This is done in the next p r o p o s i t i o n . 2.7. Proposition : E-B i m p r i m i t i v i t y

Let H = GO, B = Cc(GO) and E = Cc(G2,a). Then X = Cc(G) is an bimodule. In other words, C ~ (G2,~) and C* (GO) are s t r o n g l y Morita

equivalent. Proof :

I t is not s u r p r i s i n g t h a t t h i s r e s u l t is independent of { since G2, being

c o n t i n u o u s l y s i m i l a r to G0, has t r i v i a l

cohomology. Thus we may assume t h a t ~

have to check the l a s t c o n t i t i o n s in the d e f i n i t i o n B-valued inner-product is c l e a r l y p o s i t i v e

of an i m p r i m i t i v i t y

1. We

bimodule. The

:

B(U) = J 1 f ( y - 1 ) I 2 d ~ U ( y ) . The E-valued inner-product is p o s i t i v e

; as mentioned before, we can f i n d here an

approximate i d e n t i t y f o r the r i g h t action of Cc(G ) of the form B ; namely, l e t K be a compact subset of GO, g ~ Cc(G ) nonzero on K and

h eCc(G O) such that

h(u) = I l l g(y-1)12d~'U]-l/2 f o r u E K ; then set fK = gh. To complete the proof we need only v e r i f y the norm conditions E ~ IIbll2

E and <ef,ef> B ~ Nell2 < f ' f > B

where e E E, b e B and f ~ X. But

81 E(x,y ) = Ibor(x) I2

Hbll2 E -

E(x,y) and

E = m

where c(x) = (IImll2 - I b o r ( x ) 1 2 ) l / 2 .

Assume t h a t e c E is non-negative. Then

<ef,ef>B(U ) = J l e f ( Y -m) Imd~u(y) = I lle(y,y-lz)l/2f(z-1)e(y,y-lz)l/2dxr(Y)(z)12dxU(Y)

_
le(y,z)d~d(Y)(z)#

I

r

d~,U(y)

e ( y , y - l z ) I f ( z - I ) 12dXr ( z ) (y) dxU(z)

~le~Ii,r (sup ye(zy,y-1)d~d(Z)(y))

<_

Y'e(y,y ~z)d~, (Y)(z) J If(z -I) i2d>,U(z)

Z

!lelli, r IIeEII,dB (u)

_<

< ,el I

B(U).

This gives a .-homomorphism L : E ~ L(X) where L(X) is the algebra of bounded operators on the pre B - H i l b e r t space X which is bounded when E has the IT Ill-norm ; therefore, i t is normidecreasing. By d e f i n i t i o n of the C* -norm on E, IIL(e)II < llei! and hence the required i n e q u a l i t y . OE.D. A representation of C* (GO) can be induced up to a representation of C*(G2,~) by R i e f f e l ' s tensor product construction (Corollary 6.15 of [ 6 ~ )

and " r e s t r i c t e d "

to C * (G,o), which acts on C * ( G2,~ ) as double c e n t r a l i z e r s (a function on G can be viewed as a function on G2

depending on the second v a r i a b l e only). A l t e r n a t i v e l y ,

the

r e s t r i c t i o n map P : Cc(G,m ) ÷ Cc(GO) is a generalized conditional expectation ( [ 6 3 ] , d e f i n i t i o n 4.12) and so a representation of C* (GO) may be induced to C * (G,o) via P. Let us construct e x p l i c i t l y will

these representations induced from the u n i t space ( t h i s

be used in 3.2). For s i m p l i c i t y ,

consider a m u l t i p l i c i t y - f r e e

representation of

C * (G0) , given by m u l t i p l i c a t i o n on the space L2(G 0 ,~), where p is a measure on GO. The space of the induced representation is obtained by completing Cc(G ) ®Cc~O)Cc(Q O) = Cc(G ) with respect to the inner product


.

h, g

,

k>: I P(g* * f)h = ~f

®

h){g

: I ®

k]dv -I

u)

82 where, as usual, v =jdxUd;~(u). The induced representation, denoted by !ndp L2(G,v -1) by convolution on the l e f t

acts on

: for f E Cc(G) and ~,q s L2(G,v -1)

(Ind~(f)~,~) = f f ( x y ) ~ ( y - m ) ~ ( x ) ~ ( x y , y " I ) d~U(y)dXu(X)d~(u). We have met this representation before, in the case when ~ is a quasi-invariant measure (proposition 1.10) : i t is the regular representation on

~z.

I t results from proposition 1.11 that the function defined by NfiIre d = sup IIL(f)II, where L ranges over all representations induced from the unit space, is a C -norm on Cc(G,~ ) dominated by the 2.8. Definition

C*

-norm Nfil.

: The reduced C* -algebra Cred(G,~ ) of G is the completion of

Cc(G,o ) for the reduced norm [I fire d. I t is a quotient of C*(G,~) since the i d e n t i t y map on Cc(G,~ ) extends to a • -homomorphism of C (G,~) onto Cred(G,c~). Representations induced from more general subgroupoids w i l l only be considered in the context where theorem 1.21 applies. The notion of generalized conditional expectation used the following proposition was introduced by M.Rieffel in [6~ ( d e f i n i t i o n 4.12). This is the piece of structure which allows the construction of induced representations. 2.9. Proposition :

Assume that G is second contable and that G and H have s u f f i c i e n -

t l y many non-singular Borel G-sets. Then the r e s t r i c t i o n map from the pre-C*-algebra Cc(G,~ ) to the pre-C -algebra Cc(H,o ) is a generalized conditional

expectation.

The following lemma shows the p o s i t i v i t y of P ; i t is due to Blattner in the case of groups 2.10. Lemma :

([~,

theorem 1, page 424). We follow here [ 6 ~ ,

theorem 4.4.

Let (u,~,L) be a 1-representation of H ; then, for any f,g e Cc(G,1 )

and any ~,~ ~ I~(~) (the space of square-integrable sections), we have ( L o p ( g * * f)~,n) = ~ b ( x ) ( ~ ( f , ~ ) ( x ) , ~ ( g , ~ ) ( x ) ) d v ( x )

where b is a Bruhat approximate

cross-section for G over H\G, v =5~u d~(u) and ~(f,~)(x)

=

f(x'lk)L(k)~°d(k)

of u r e l a t i v e to ( H , ~ ) . )

AHI/2(k) d ~ (z)(k)"

(&H denotes the modular function

83 Proof : We have (Lop(g* . f)g,q) = J g* * f(h) (L(h)

god(h), nor(h)) dVHo(h)

where vHO is the symmetric measure of ~ relative to (H, x~),. This, in turn, equals / / # ( x - l h -1)

f(x -1) (m(h) god(h), nor(h)) dxd(h)(x) dvmo(h)

=J/(Jb(k-lx) d~ (x) (k))

g(x-lh -1 f(x -1) (L(h) god(h), nor(h))

=Jf~b(k-lx) ~(x-lh -1) f(x -1) (L(h)

dxd(h)(x) dvmo(h)

~od(h),nor(h)) dxr(k)(x) d~(h)(k) dvmo(h )

The use of Fubini's theorem is legitimate, because the support of the function (k,x)÷b(k-lx)

g(x-lh -1) f(x -I) is compact :

k-1 x E H suppf~n suppb. We make the change of variable x ~ kx in the last integral to obtain

Jffb(x) g(×-lk-lh-1) f(×-lk-1) (L(h)~od(h), qor(h))d~d(h)(×)dx~(h)(k)dVho(h). .1/2 (h)d~Hud~(u) and change the order of integration ; this is !{e write dVHo(h ) = f~H justified as above. We get J b(x) ~(x-lk-lh -1) f(x-lk -I) (L(h)Cod(h),qor(h))

A~/2(h) d~Hr(x)(h)

dxd(k)(x) dX~(k) dp(u) . We make the change of variable h ~ hk -1, .yielding I b(x) ~(x-lh -1) dXad(k)(h) We

f(x-lk -1) (L(hk -1) ~or(k),qor(h)) A~/2 (hk -1) dxd(k)(x) dvH(k ).

use the fact that AHI(k) dvH(k ) = dvHl(k) to produce b(x) g(x-lh -1) f(x-lk -1) (L(k-1)¢or(k),L(h-1)nor(h))

A~/2(hk)

d~Hu(h ) dxU(x) d~Hu(k ) d~(u). Finally, we change the order of integration and arrive at .1/2 (k), g(x- lh-1 ) fb(x) (~(f(x-lk-1)L(k-1)~°r(k) ~H L(h-l)nor(h) A~/2(h))

dXHu(h) d~Hu(k))dxU(x ) dp(u) .

The vector-valued function k ~ f(x-lk -I) L(k-1)~or(k)

&~/2(k) is ~Hr(x) integrable

because it is measurable and its norm is integrable since l~or(k)[ A~/2(k) is locally integrable. Hence ~(f,~)(x) =~f(x-mk -1) m(k-1)~or(k) 5~/2(k) d~Hr(x)(k ) makes sense and (r(f,~)(x),

v(g,n)(X)) is equal to

(f(x-lk-1)m(k-m)gor(k)

A~/2(k), g(x-mh-1)L(h-1)nor(h) A.1/2~ H kh)) dIHr(x)(h)dXHr(x)(k). Q.E.D.

84 Proof of the p r o p o s i t i o n . (i)

P is s e l f a d j o i n t ,

because P ( f * )

= P(f)*

f o r f c Cc(G,o ) by simple

calculation. (ii)

P is p o s i t i v e ,

we f i r s t

i.e.,

consider the case

P(f**f)

! 0 in C (H,o) f o r f s Cc(G,o ). To see t h i s ,

g = I. The lemma t e l l s

us t h a t L ( P ( f * * f ) ) ~ 0 f o r any

representation L of C (G,1) obtained by i n t e g r a t i o n . S i n c e C*(G) has a f a i t h f u l

family

of such representations by theorem 1.21, P ( f * * f ) Z O. To deal with the case of an arbitrary of G°

cocycle o, we consider instead G° and H° . Since Ha is a closed subgroupoid

and G° and H° s a t i s f y the hypothesis of the p r o p o s i t i o n ,

from Cc(G~,I) onto Cc(H°,I) is p o s i t i v e . Cc(Ge,I)

Q

the r e s t r i c t i o n

map Q

Since the diagram

~ Cc H~,I)

with

I

~f(x) =

Sf(s,x)sds

~mg(h ) = ] g ( s , h ) s d s ~H

Cc(G,~ )

P

• Cc(H,~ )

commutes and since ~ is onto while ~H is continuous, P is also p o s i t i v e . (iii)

P satisfies

the expectation property ; i . e . ,

P(¢-f) = # * P ( f )

f o r f s Cc(G,o)

and # s Cc(H,o ), as can be seen immediately from the d e f i n i t o n of ¢ - f . ( i v ) P is r e l a t i v e l y

bounded ; t h a t i s , f o r every g s Cc(G), the map

f ~ P(g* , f , g) is bounded with respect to the C * - n o r m s ) . in p r o p o s i t i o n 4.10 of [63]. F i r s t , P(f* * g * * g * f) ~

To see t h i s ,

proceed as

one establishes the i n e q u a l i t y

llgll2 P ( f * *

f) .

I t s u f f i c e s to show the i n e q u a l i t y when one evaluates both sides against a p o s i t i v e type measure, t h a t i s , a measure v on Cc(H ) s a t i s f y i n g v ( f ~ f )

~ 0 f o r any f s Cc(H).

Via the GNS c o n s t r u c t i o n , a p o s i t i v e type measure on G defines a representation of Cc(G,o), hence of Cc(G,o ) by theorem 1.20 ; in p a r t i c u l a r ,

i t is continuous w i t h

respect to the C -topol O ~,, v°P is a p o s i t i v e type measure on Cc(G,o) ~ j . Therefore, The corresponding representation is given by convolution on the l e f t '~ IIg * fll~op ~ Ilgll m I]fll ~ op where ( , )voP vop((g

. f)

;

f o r g , f s Cc(G),

is the inner-product defined by voP : .

(g . f ) )

~

llgll2 voP (f** f) .

Then, one applies the generalized Cauchy-Schwarz i n e q u a l i t y of p r o p o s i t i o n 2.9 of [63]

85 to conclude. (v) Let (ek) be an approximate l e f t

i d e n t i t y f o r Cc(G,o ) with the i n d u c t i v e l i m i t

topology. Then, f o r any f e Cc(G,~ ), ( f - e k , f ) * . and i t s r e s t r i c t i o n

( f - e k . f ) tends to 0 in Cc(G,~ )

to H tends to 0 in Cc(H,~ ), hence in the C*-norm.

( v i ) The range o f P is Cc(H,~ )(vii)

P is c l e a r l y f a i t h f u l

; f**

f = 0 on H ~ > f * *

f = 0 on GO ~ > f = O. Q.E.D.

This p r o p o s i t i o n allows a p a r t i a l answer to a question t h a t has been avoided u n t i l now. Given a l o c a l l y compact groupoid, we have assumed the existence o f a Haar system and kept i t f i x e d . Most notions introduced, such as q u a s i - i n v a r i a n c e or the c o n v o l u t i o n product, depend e x p l i c i t l y

on the choice of such a Haar system. What is

the r o l e of t h i s choice and can we f i n d notions independent of i t 2.11. C o r o l l a r y :

?

Let G be a second countable l o c a l l y compact groupoid, ( ~ )

two Haar systems w i t h respect to which G has s u f f i c i e n t l y

i = 1,2

many n o n - s i n g u l a r Borel

G-sets and l e t ~ be a continuous 2-cocycle. Then, the corresponding C * - a l g e b r a s C (Gl,a) and C (G2,~) are s t r o n g l y Morita e q u i v a l e n t . Proof :

We set G = G1 and view G2 as the subgroup H. Then H\G2 = G1. We can use

p r o p o s i t i o n 2.9 to show t h a t X = Cc(G) is indeed an E:B i m p r i m i t i v i t y

bimodule with

E = Cc(H\G2,~ ) and B = Cc(H,~ ) as before. Propositon 2.9 gives the p o s i t i v i t y B as well as the norm c o n d i t i o n <ef,ef> B ~llell I

of

B f o r e , f E Cc(G). By

symmetry, s i m i l a r statements hold f o r E. Q.E.D. 2.12. Example : (i.3.28.c)

Let X be a second countable l o c a l l y compact space. We have defined

the t r a n s i t i v e groupoid on the space X as G = X x X, with the groupoid

s t r u c t u r e given in 1.1.2 ( i i )

and the product topology. We know t h a t a Haar system

on G is defined by a measure ~ o f support X. I f X is uncountable and m is non-atomic, then G has s u f f i c i e n t l y

many n o n - s i n g u l a r Borel G-sets. Let us f i x m. Since the class

of m is the only i n v a r i a n t measure class and any r e p r e s e n t a t i o n of G is a m u l t i p l e

86

of the one-dimensional t r i v i a l

representation, the corresponding C * - a l g e b r a is iso-

morphic to the algebra of compact operators on a separable H i l b e r t space. Thus two measures ~1 and ~2 give isomorphic C*-algebras but there is no canonical way to construct an isomorphism. I t would be i n t e r e s t i n g to have an example where two Haar systems give non-isomorphic C * - a l g e b r a s .

3. Amenable Groupoids

The notion of amenability for groups (see [41] or [30]) takes many forms and a large part of the theory consists in showing t h e i r equivalence. Our goal is much more l i m i t e d here. We shall f i r s t

consider measure groupoids and choose a d e f i n i t i o n of

amenability best suited to our needs. We seek a condition ensuring that every representation is weakly contained in the regular representation. Then the von Neumann algebra associated to any representation is i n j e c t i v e ; here, the proof is e s s e n t i a l l y the same as in [83], where R.Zimmer studied ergodic actions of countable discrete groups. A notion of amenability is then given for l o c a l l y compact groupoids with Haar system, whose main advantage is that i t is e a s i l y checked.Some examples are studied. Throughcut t h i s section, G designates a l o c a l l y compact groupoid with a f i x e d Haar system {~u}. 3.1. D e f i n i t i o n : A q u a s i - i n v a r i a n t p r o b a b i l i t y measure ~ on GO w i l l be called amenable (we also say that

(G,u) is amenable) i f there exists a net ( f i )

in Co(G)

such that (i)

the functions u ~ S l f i l 2 d ~ u converge to 1 in the weak * - t o p o l o g y of

L~(GO,~) and (ii)

the functions x ~ f f i ( x Y )

-topology

of

L~(G,~) where v

~ i ( Y ) d x d ( X ) ( y ) converge to 1 in the weak

.

is the induced measure of p.

This d e f i n i t i o n reduces to one of the equivalent d e f i n i t i o n s of amenability in the case of a group ; namely, that the function i is the l i m i t ,

uniformly on compact

sets, of functions of the form f . f % where f ~ Cc(G)(one has to use theorem 13.5.2 of [19]).

87 The t r a n s i t i v e

measures on a principal

measures. Let us indicate b r i e f l y ~u] and l e t u be the t r a n s i t i v e of compact sets in ~ ]

groupoid provide examples of amenable

how the net ( f i )

can be constructed.

Fix an o r b i t

measure d . ~u . One can choose an increasino net (Ki)

(with the topology given by the b i j e c t i o n d : Gu ÷ ~ ] )

such

that uKi = [u]. Define f i by fi(x)

= ~(Ki )-1/2 i f

(r,d)(x)

s Ki

x Ki,

= 0 otherwise. Then f .1* f~(x) = 1 i f

(r,d)(x)

s Ki

x Ki

= 0 otherwise. The function f i is not in Cc(G ) but i t is in L2(9,v) (where u has been normalized) and i t can be approximated in L2(G,v) by elements of Cc(G ). 3.2. Proposition

: Let u be a q u a s i - i n v a r i a n t

amenable p r o b a l i l i t y

measure on GO and

o a 2-cocycle in Z2(G,~). Then the integrated form of any a-representation

of G

l i v i n g on u is weakly contained in the regular representation on u of C*(G,a). Proof :

We follow Takai ([70],

page 29). Let ( U , ~ u , L )

A vector state of the integrated representation ~(f) = f f ( y )

(L(y)~od(y),~or(y))

By 3.1 ( i i ) ,

of G.

is of the form

d~o(y ) f o r f s Cc(G )

where ~ is a u n i t vector in F(~Q. Let ( f i ) @i(f) = S ( S f i ( x ) - ~ i ( y - l x ) d ~ r ( Y ) ( x ) )

be a a-representation

be as in 3.1 and define @i(f) by f(y)

(k(y)~od(y),~or(y))dvo(Y).

@i(f) tends to @(f). Moreover, a routine computation allows us to w r i t e

the equation ~i(f)

=~f(xy)

(~i(y-1),

~ i ( x ) ) a(xy,y -1) d~U(y) d~u(X ) d~(u) ,

where ~i(x) is defined by ~ i ( x ) = Dl/2(x) ~ ( x , x - 1 ) T i ( x ) k ( x - Z ) ~ o r ( x ) . We recognize the expression for @i(f) as ( I n d M ( f ) ~ i , C i ) , tation induced by the r e s t r i c t i o n space r(I~) of square-integrable

where IndM is the represen-

M of L to C*(G O) (see end of 2.7).

sections of the H i l b e r t bundle~(~u = L2(G,~u) ® ~

(GO,u). Let us compute the norm of Ci" II~ ill 2 = ~II~i(x)II 2 d v - l ( x ) =;[fi(x)I2[kor(x)II

I t acts on the

2 du(x) (because D -

dv

d~-I

)

on

88 = ]ll~(u)ll 2 ( I f i ( x ) l

II~ill

By 3.1 ( i ) ,

obtain l { ( f ) l

2 dxU(x)) d p ( u ) .

tends to 1. From the inequality l ~ i ( f ) I L

~ lllndM(f)ll.

lllndM(f)N

II(il I

ll~iIl,

we

Since IndM is a direct integral of representations equi-

valent to the regular representation on # of C (G,~), i t is weakly contained in i t and so is L. Q.E.D. 3.3. Remark : One expects a converse ; namely, i f the integrated form of the t r i v i a l one-dimensional representation of G l i v i n g on u is weakly contained in the regular representation on ~ of C*(G), then ~ is amenable. Let us say that a continuous function # on

G is of positive type (with respect to u) i f ~ ( f )

=If(x)

~(x) dvo(X ),

f E Cc(G), defines a positive linear functional ~ on C~(G). For example, the function 1, which is associated with the vector state ~ ( f ) trivial

=~f(x)dvo(X ) of the one-dimensional

representation (~,~u u = ~, L = I) is of positive type. Let us determine the

positive type functions associated to the vector states of the regular representation ( ~ ' ~ u = L2(G'~U)' L(x))where L(x)~(y) = ~ ( x - l y ) . The vector ~ ~ Cc(G) c L2(G,v) gives the positive type function (k(x) ~od(x),~or(x)) = ~ ( x - Z y )

~(y)d~r(X)(y)

= ~ . ~*(x-1).

Hence, i f our hypothesis holds, the state ~1 is a weak l i m i t of states associated with positive type functions which are f i n i t e sums of functions of the form

~,~*(x-1),

with ~ C c ( G ). I t is not hard to show that these positive type functions can in fact be chosen to be of the form ~ . ~ * ( x - 1 ) .

Indeed one observes that for ~, f,g c Cc(G),

w~(f * g*) =J~(x) f . g* (x) dvo(X ) = J ~ ( x ) f . g* (x -1) dvo(X ) =

(where ~(x) = ~(x-1))

(L(~)f,g)

Hence, i f ~ is of positive type, L(~) is a positive operator. Then, using Kaplansky's density theorem to approximate i t s square root, one obtains a net ( f i ) that L(f i . f~) ÷ L ( ~ ) i n

the weak operator topology and L ( f i . f ~ )

that the positive linear functionals associated to f i * f *i

in Cc(G) such

~ L(~). This implies

(x -1) converge weakly to

89 m@. To conclude, one would need to e x h i b i t ~1 as a weak l i m i t

of states associated

with p o s i t i v e type functions of the form f i * f *i (x -1) with f i e Cc(G) which are uniformly bounded in L~(G,v). So far I have been unable to do t h i s . 3,4. Lemma : amenable i f f

Let ~ be a q u a s i - i n v a r i a n t

probability

measure on G0, Then u is

there e x i s t s an approximate i n v a r i a n t mean on

(gi) of non-negative functions

L~(G,v), that i s , a net

in Cc(G ) such that

( i ) the functions u ~jgid~ u converge to 1 in the weak . - t o p o l o g y of L~(GO,~) ; and (ii)

the function x ~ i g i ( x Y

) - gi(Y)Id~d(X)(y)

converge to 0 in the weak

*-topology of L=(G,v). Proof : The proof is e s s e n t i a l l y 61). Let us s t a r t with ( f i )

the same as in the case of a group (e.g.

as in 3.1 and define gi = I f i 12" The f i r s t

immediate. Using the i n e q u a l i t y

llal 2 - !b12~< (lal

+ Ibl)(la

[41], page

property is

- bl) and Cauchy-Schwarz,

one obtains J Igi(xY) - gi(Y)Id~d(X)(y)

< [ S ( I f i ( x y ) I + I f ~ ( y ) I ) 2 d ~ d ( X ) ( y ) ] 1/2 [ f l f i(xy) - fi(y)Imd~d(x) (y)]1/2 .

Let us set hi(u ) = I I f i ( y ) I 2 d ~ U ( y ) .

The f i r s t

member of the product is majorized by

2 I/2 [hi or(x ) + h i o d ( x ) ] i / 2 while the second is majorized by [11 " hi°r(x)l

+ 11 - hi°d(x)I

+ I1 - fi

+ 11 - fi*

f l (x)I

* f T ( x - 1 ) l ] 1/2

The s-topology on L~(G,~) is defined by the semi-norms ~@(f) = (f@tf[2dv) 1/2 where ~p is a non-negative element of LI(G,~).

The f i r s t

term goes to 2 in the s-topology

and is bounded in the L~(G,.~) norm and the second goes to O. Thus t h e i r product goes to 0 in the s-topology and a f o r t i o r i

in the weak * - t o p o l o g y .

with ( g i ) , we define f i = gi 1/2. Again, the f i r s t satisfied.

Using the i n e q u a l i t y

Conversely, s t a r t i n g

property of ~. R i is immediately

la - bl 2 _< la 2 - b21, one obtains without much trou-

ble the estimate I1 - f f i ( x Y ) f - ( y ) d ~ d ( X ) ( y ) l

_<

1/2 [ J l g i ( x Y ) - g i ( Y ) I d ~ d ( X ) ( y ) +

Ii-

Sgi(Y)d~r(X)(y)I

+ 11 _ i g i ( Y ) d ~ d ( X ) ( y ) l

].

Q.E.D.

90 3.5. Proposition : Let ~ be a q u a s i - i n v a r i a n t amenable p r o b a b i l i t y measure on GO and o a 2-cocycle in Z2(G,~). Any a - r e p r e s e n t a t i o n of G l i v i n g on ~ generates an i n j e c t i v e yon Neumann algebra. Proof : As mentioned e a r l i e r , a m e n a b i l i t y we use - i t (~,~u,L)

the idea of the proof is i n Zimmer [83]. The notion of

is more s t r i n g e n t than Zimmer's - makes the proof easier. Let

be a a - r e p r e s e n t a t i o n of G ; L also denotes the integrated representation

on F(}6) given by ( k ( f ) ~ , n )

= ~ f(x)(L(x)~od(x),nor(x))d~o(X)

JK~ denotes the von Neumann generated by { L ( f )

~,n c r(~6)

f E Cc(G) ;

• f s Cc(G)}; J~6' is i t s commutant and

~) is the algebra of decomposable operators on F(~z~). An operator A E ~ acts on F ( ~ ) by A~(u) = A(u)$(u) where A(u) is an operator on ~ . {A ~ )

: Aor(x) L(x) = L(x) Aod(x) f o r

Neumann algebra is i n j e c t i v e

iff

We note t h a t ~

v a.e. x}. Tomiyama has shown t h a t a v o n

i t s commutant is i n j e c t i v e

; in p a r t i c u l a r ~), which

is the commutant of a commutative von Neumann algebra, is i n j e c t i v e . a c o n d i t i o n a l expectation of 5Donto d~6' ; t h i s w i l l injective.

We w i l l

construct

show t h a t ~6', hence J~6, is

Let ( g i ) be a net as in 3.4 and l e t M be a bound f o r sup S g i d~u" We u

define a l i n e a r map Pi : ~ ) ~ ) PiB(u) = S g i ( x ) i.e.

=

by

k(x) Bod(x) k(x)'Zd~U(x)

(PiB~,n) = f g i ( x )

(L(x)Bod(x)k(x)-l$or(x),nor(x))dv(x)

f o r ~,n ~ r ( ~ ) . There is no problem checking t h a t Pi is well defined. Horeover since IIPiB(u)II IIPiBIl

<

IIBII ~ g i ( x ) d ~ U ( x ) , we see t h a t

< MIIBII.

We also note t h a t Pi is p o s i t i v e . The P i ' s are u n i f o r m l y bounded i n norm. Hence there is a subset converging to a bounded p o s i t i v e l i n e a r map P i n the f o l l o w i n g sense. For every p a i r of vectors (~,n) i n F ( ~ ) The r e s t r i c t i o n

and f o r every B i n ~), (PiB~,n) tends to (PB~,n).

of P to JK~' is the i d e n t i t y

(in particular,

A E ~ , then (PiA~,n) = f g i ( x ) = By 3.4.

(i),

(Aor(x) ~ o r ( x ) , n o r ( x ) ) d v ( x )

J(A(u)~(u),n(u))(Igi(x)dxU(x))du(u)

we obtain at the l i m i t ,

P is u n i t a l ) .

For i f

91 (PAd,n) = j ( A ( u ) ~ ( u ) , n ( u ) ) d ~ ( u )

=

(A~,n).

The proof that P is an expectation w i l l

be completed when we show that P(~)) = ~ .

A f t e r routine computations, one obtains, for B e ~ for f ~ C~(G) and ~,n ~ r ( ~ ) , (L(f)PiB~,n)

=

~f(x)gi(y)~(x,y)~(y,y-m)(m(xy)Bod(y)m(y-m)~od(x),nor(x))dxd(X)(y) ((PiB)L(f)~,n)

d~o(X ), and

=

ff(x)gi(xY)o(y-lx-l,x)~(xy,y-lx-1)(L(xy)Bod(y)k(y-1)~d(x),nor(x))

dxd(X)(y) d~o(X ).

One notes that o ( x , y ) ~ ( y , y -1) = ~ ( y - l x - l , x ) j ( x y , y - l x - Z ) . Hence the following estimate holds: [((L(f)PiB

- PiBL(f))~,n)1

[IBH f I f ( x ) l

llnor(x)ll

H~°d(x)II

;]gi(xY)

- gi(Y)l

dxd(X)(Y) duo(X)-

Since jlf(x)t

lt~°d(x)![

we may use 3.4 ( i i )

Nn~r(x)lt

dvo(X)~

tlfili

!t~11 llhlf,

to conclude that the r i g h t hand side goes to zero.

Hence L(f) PB = (PB)L(f) and PB E ~ ' . Q.E.D. 3.5. Remarks : (a) R.Zimmer has introduced in [ 8 ~ , mean f o r (G,~).

definition

I t is a p o s i t i v e unital

4.1, the following notion of i n v a r i a n t

l i n e a r map m from L~°(G,~) onto L~(GO,~)

satisfying ( i ) m(h¢) = hm(¢) f o r ¢~L~(G,v) (ii)

and h s Cc(GO), where he(x) = h o r ( x ) ¢ ( x ) ,

and

m(f¢) = fm(¢) f o r ¢ s L<(G,v), and f s Cc(G ), where f¢(x) = J f ( y ) ¢ ( y - l x ) d ~ r ( x ) ( y )

and where f@(u) = f(y)~od(y)dxU(y)

for ¢ s

L(GO,u). By a compactness argument, the

existence of an approximate i n v a r i a n t mean as in 3.4 gives the existence of an i n v a r i a n t mean. The converse is probably true, but I don't have a correct proof.

I t can be shown

as in [82] and [83], where the case of an ergodic action of a countable discrete group is considered, that for a discrete groupoid G, the regular representation on u generates an injective

yon Neumann algebra i f f

there is an i n v a r i a n t mean f o r (G,u).

92

(b) R. Zimmer has also defined in [ 8 ~ , d e f i n i t i o n 1.4, an amenable ergodic group action by a f i x e d point property. This property is equivalent to the existence of an i n v a r i a n t mean in the discrete case ( [ 8 ~ ,

4.1) but the general case is unknown.

The d e f i n i t i o n of amenability given in 3.4 implies the f i x e d point property, as a standard averaging process shows. (c) One can also use the approximate i n v a r i a n t mean of 3.4 to average cocycles and get a vanishing theorem (cf. Johnson, [48], 2.5, page 32). 3.6. D e f i n i t i o n : Let us say that G is measurewise amenable i f every q u a s i - i n v a r i a n t measure on GO is amenable. I f a l l the representations of C*(G,o) are obtained by i n t e g r a t i o n and i f G is measurewise amenable, i t results from 3.2 that C*(G,~) coincides with the reduced C * - a l g e b r a Cr~d(G,~ ) and from 3.5 that i t is nuclear. A s u f f i c i e n t condition for G to be measurewise amenable is the existence of a net ( f i )

in Cc(G) such that

( i ) the functions u ~ S Ifi(x)12d~U(x) are uniformly bounded in the sup-norm ; and (ii)

the functions x ~ f f i ( x Y ) f i ( Y ) d ~ d ( X ) ( y ) converge to I uniformly on any

compact subset of G. This condition is also necessary in the case of a group (cf. Dixmier [19], 13.5.2, page 260) ; but I do not know i f i t is true in general. Since t h i s condition is handy, I call i t amenability, although I don't have any real j u s t i fication for it. A question which arises is the amenability of C (G,~) in the sense of Johnson ( [ 4 8 ] , 5, page 60) ; in p a r t i c u l a r , does the above condition imply amenability ? Let us now look at how amenability is preserved under some operations. 3.7. Proposition :

Let U be a l o c a l l y closed subset of the u n i t space of G. I f G

is [measure wise]amenable, the reduction GU is [measure wise] amenable. Proof :

Suppose G amenable. Then there e x i s t s a net ( f i )

in Cc(G) such that f i * f i

converges to I uniformly on the compact sets of G and I f i . f ~ ( u ) I

< M f o r suitable

M and any u. Let ( h i ) be an approximate i d e n t i t y on Cc(U ), bounded in sup-norm. Then

93 gi(x) : hior(x ) fi(x)

hiod(x ), x c GU

defines a net in Cc(Gu) s a t i s f y i n g the required p r o p e r t i e s . The proof of the other statement is s i m i l a r . We note t h a t any restriction

to U o f a q u a s i - i n v a r i a n t measure on GO, namely, a q u a s i - i n v a r i a n t measure

on U is e q u i v a l e n t to the r e s t r i c t i o n G (1.3.7). respect to

q u a s i - i n v a r i a n t on U is e q u i v a l e n t to the

We denote the r e s t r i c t i o n {~}

[~](E) = 0

of the s a t u r a t i o n

[u] of

~ w i t h respect to

of ~u to GU by ~ and the induced measure w i t h

by v U = %xuU d~(u). Then f o r E c U, iff

v(d-l(E)) = 0 ;

iff

for

u a.e. u,

xU(d'l(E)) = 0 ;

iff

for

p a.e. u,

x~(d-l(E))

= 0 (because E

U and ~ l i v e s

on U) ; iff

vu(d-l(E)) = 0 ;

iff

v u l ( d - l ( E ) ) - = 0 (because ~ is q u a s i - i n v a r i a n t )

iff

~(E)

=

0

;

.

Q.E.D. 3.8. Proposition :

Let G be a l o c a l l y compact groupoid w i t h Haar system, l e t A be

a l o c a l l y compact group and c a

continuous l - c o c y c l e in ZI(G,A). Let G(c) be t h e i r

skew product ( 1 . 1 . 6 ) . (i) (ii)

I f G is [measurewise]

amenable, then G(c) is [measurewise] amenable.

I f A is amenable and G(c) is [measurewis~

amenable, then G is [measurewis~

amenable. Proof : Let us r e c a l l the d e f i n i t i o n

1.1.6 of G(c) : G(c) = G × A w i t h

( x , a ) ( y , a c ( x ) ) = ( x y , a ) and (x,a) - I = ( x - l , a c ( x ) ) .

I t s u n i t space is GO × A. The

l o c a l l y compact groupoid G(c) has been defined before 1.4.10. I f { u} is a Haar system f o r G, a Haar system { u,a} f o r G(c) is given by ff(x,b)d~u'a(x,b)

= ~ f(x,a)d~U(x)

.

Let us describe q u a s i - i n v a r i a n t measures f o r G(c). Suppose t h a t ~ is a q u a s i - i n v a r i a n t measure on GO f o r G and {mu} a system of measures on A which is ~adequate (Bourbaki [613.1) ( t h i s means t h a t p = f ~ud_~(u) is well defined) and which s a t i s f i e s ~d(x) ~ ~ r ( x ) c(x) f o r ~ a . e ,

x, w h e r e ~ i s the induced measure on ~. Then ~ = I ~ u dp(u)

94 is a q u a s i - i n v a r i a n t

measure for G(c). Conversely, i f the q u a s i - i n v a r i a n t

can be disintegrated

along the f i r s t

measure

projection of GO × A, i t is of that form. For

the proof of this f a c t , we may assume that ~,~ and ~ a.e. mu are p r o b a b i l i t y measures and we may replace {~u} by equivalent p r o b a b i l i t y

measu'res. The measure v induced by

is of the form S fdv while - 1

= ~f(x,a)d~U(x)d~u(a)d~(u), is of the form

fdv -1 = ~ f ( x - l , a c ( x ) ) d ~ U ( x ) d m u ( a ) d ~ ( u ) In p a r t i c u l a r ,

.

for any measurable set E in C, v(E x A) = ~(E) and ~-I(E × A) = ± - I ( E ) .

This shows that ~ is q u a s i - i n v a r i a n t . of v along the f i r s t

Then, using the uniqueness of the d i s i n t e g r a t i o n

projection of G × A, one gets mr(x) ~

~d(x) c(x-1) for v a . e . x .

Conversely, there is no problem checking that a measure ~ of the above form is quasiinvariant. ( i ) Let u = ~ u d ~ ( u )

be a q u a s i - i n v a r i a n t

amenable, there exists a net ( f i ) weakly . in L=(GO,~), and x ÷ f i be an approximate i d e n t i t y

measure for G(c) as above. I f ~ is

in Cc(G ) such that u -~ f i * f#(u) converges to 1 * f~(x) converges to I w e a k l y * i n

for Cc(A ) with pointwise m u l t i p l i c a t i o n

L~(G,~). Let (hi) and bounded in

sup-norm and define gi ~ Cc(G × A) by gi(x,a)

= f i ( x ) h i ( a ).

The net (gi) has the required properties gi * g~(x,a) = ~ g i ( x y , a ) g i ( Y , a c ( x ) )

: d~d(X)(y)

= hi(a) hi (ac(x)) f i * f i (x)Let us check the convergence of (u,a) ÷ g i * gi (u'a)" The net is bounded in L~(G 0 × A,u), hence i t is enough to check the convergence against functions of the form f ( u ) g ( a )

where f ~ Cc(G° ) and g ~ Cc(A ).

We see that f(u)g(a) =~f(u)

gi * g i (u'a) d~(u,a)

([g(a)lhi(a)l

2 d~u(a)) f i * f ~ ( u )

d~(u) goes to

f(u) g(a) du(u,a), Since ~ g ( a ) l h i ( a ) l

2 dmu(a ) goes to ~g(a) dmu(a ) in LI(GO,~) and f i * f i (u) goes to

I in (L~(GO,u), weak.), The convergence of g i * g i

(x'a) is proved in the same fashion.

95 This shows that ~ is amenable. One proves in the same way that i f G is amenable, then G(c) is amenable. (ii)

We assume that A is amenable and ~(c)

is

measurewise amenable. Let ~ be a quasi-invariant

measure on GO. Then ~ = L x ~,

where m is a r i g h t Haar measure for A, is quasi-invariant

for G(c). Since G(c) is

amenable, there exists an approximate invariant mean ( g i ) , gi ~ O, gi ~ Cc(G × A) such that (u,a) + ~ gi(x,a)d~U(x) converges to i in (L~(G 0 x A,~), weak*), and (x,a) ÷ } Igi(xy,a) - gi(Y,ac(x) l

d~d(X)(y) converges to 0 in (L~(G x A,v), weak*).

group A also has an approximate invariant mean (ki) S kj(a)d~(a)

= i , and b ~ f l k j ( a b )

: k i ~ 0 k i e Cc(A ) such that

- ku(a)I dm(a) converges to 0 uniformly on the

compact subsets of A. Let us define f i j

e Cc(C) by f i j ( x )

=fgi(x,a)kj(a)dm(a).

is not hard to check that the family of functions u ~ f f i j ( x ) d X U ( x ) L~(GO,~) and the family of functions x ~ J l f i j ( x y L~(G,~). We w i l l

It

is bounded in

) - f i j ( y ) Id~d(X)(Y) is bounded in

show that, given a neighborhood of 1 in (L~(GO,~), weak*),

V : {h ~L~GO,~) : where @k c Cc(GO),

l}(h(u) - i) @k(U) d~(u)[ ! ek' k : l . . . . . m},

~k > O, k=l . . . . . m, and a neighborhood of 0 in (L'(G,Z), weak.),

W : { f ~ L=(G,z) : I ~ f ( x ) ~ ( x ) there exists f i j

such that u ~ f f i j

d~(x)l ! n~

~=i . . . . . n} ,

dxu is in V and

x ÷~Ifij(xy)

is in W. Let M be a bound for the norm of the functions L~(G 0 x A). We can choose j such that, for every I I~(x)l

The

I kj(ac(x) -1) - kj(a)l

d~(a) dz(x )

- fij(y)idxd(X)(Y)

(u,a) ÷ f g i ( x , a )

dxU(x) in

~ = I . . . . . n, < o~/2M

from now on, j is kept fixed. We observe that #k(U) ( f f i j ( x )

dxU(x)) d~(u)

= f (gi(x,a) d~U(x)) ~k(U) kj(a) du(u,a) goes t o } #k(U) ku(a ) d~(u,a) =f~k(U) d~(u) as i goes to large, u ÷ ~ f i j d ~ U

is in V. Similarly,

for i s u f f i c i e n t l y

(~ Igi(xy,a) - g i ( Y , a c ( x ) ) I d ~ d ( X ) ( y ) ) = 1,...,n

~. Hence for i s u f f i c i e n t l y large, we w i l l

I~(x)Ikj(a)d~(x,a)

have

~ n~ /2

.

Writing fij(xy)

- fij(Y)

= ~(gi(xy,a)

- g i ( Y , a c ( x ) ) ) k j ( a ) d ~ ( a ) + ~ g i ( Y , a ) ( k j ( a c ( x ) -1) - kj(a)) do(a),

96 we obtain the estimate l~(~Ifij(xy ) -fij(y)Id~d(X)(y))

¢~(x) QZ(x)]

_< ~(~Igi(xy,a) - g i ( Y , a c ( x ) ) i d x d ( X ) ( y ) l ~ ( x ) I k j ( a )

dv(x,a)

+~]~(x)l

dv(x,a)

(Igi(Y,a)dxd(X)(y)) I k j ( a c ( x ) - Z ) - k j ( a ) l

_< ngo This shows that ~ is amenable. One proves in the same way that the amenability of G(c) and A implies the amenability of G. Q.E.D. Dual statements hold for the semi-direct product. 3.9~ Proposition : Let G be a l o c a l l y compact groupoid with Haar system, l e t A be a l o c a l l y compact group acting continuously on G by automorphisms leaving the Haar system invariant and l e t G x

A be t h e i r semi-direct product (1,1.7).

( i ) I f A is amenable and ~ is [measurewise] amenable, then G × A is ~easurewise] (ii)

amenable.

I f the semi-direct product G x

A is [measurewise] amenable

then G is

[measurewise] amenable. Proof :

Let us f i r s t define the semi-direct product as a l o c a l l y compact groupoid

with Haar system. We require the map from A x G into G sending (a,x) into s(a) x to be continuous. Recall (1.1.7) that G x

A is the groupoid G x A with ( x , a ) ( y , b ) =

( x ( s ( a ) y ) , ab) and (x,a) -1 = ( s ( a - 1 ) x - l , a - 1 ) . I t s unit space may be i d e n t i f i e d with GO. The product topology makes i t into a l o c a l l y compact groupoid. We say that the automorphism s of G leaves the Haar system i n v a r i a n t i f s-~ u = ~s(u) ; in other words ~ f ( s ( x ) ) d ~ S - l ( U ) ( x ) = ~f(x)d~U(x) for f c Cc(GU). I f {~u} is a Haar system for G and ~ a l e f t Haar measure for A, then {~u x ~} is a Haar system for G × A. Let us check l e f t invariance : ~f(x,a)(y,b))dxs(a-1)d(X)(y)d~(b) = ~f(x(s(a)y),ab)d~s(a-Z)d(X)(y)da(b) =~f(xy,ab)dxd(X)(y)dm(b) = ~ f(y,b)d~r(x)(y)dm(b). Since the proof of this proposition is not much d i f f e r e n t from the previous one and does not involve any d i f f i c u l t y ,

we w i l l j u s t indicate how the various approximate

97 means may be constructed. (i)

Given ( f i )

such t h a t f i * f*i -, 1 in G and "(h~) J" such t h a t h~a * h*. ÷ I on A, J

set gij(x,a)

= fi(s(a-l)x)hj(a).

Then, there e x i s t s a subnet such t h a t g i j (ii)

* g i•j ÷ 1 in G × A.

Given an approximate i n v a r i a n t mean ( g i ) f o r G ×

A, we can define an

approximate i n v a r i a n t mean f o r G by s e t t i n g fi(x)

= fgi(x,a)

d~(a). q.E.D.

3.10. Example :

A transformation group a r i s i n g from the a c t i o n of an amenable group

is always amenable but the converse is not true. Let G be a second countable l o c a l l y compact group and H a closed subgroup ; i t can be shown t h a t the transformation group H\G x G is amenable i f f

H is amenable. Hence a homomorphic image of an amenable

groupoid is not n e c e s s a r i l y amenable ; however, i t is probably true t h a t the asymptotic

range ( 1 . 4 . 3 ) of such a homomorphism is amenable ( c f . Zimmer [ 8 ~ ,

3.3).

In conclusion, l e t us ask some very basic questions. (i)

Is a closed subgroupoid of a measurewise amenable groupoid also measurewise

amenable ? This is probably true but I can prove i t only in the case of an r - d i s c r e t e groupoid. The proof uses 3.3 and 4.1. (ii) (iii)

Does 3.9 hold f o r more general extensions 7 Is amenability preserved under (the appropriate notion of) s i m i l a r i t y

?

4. The C * - A l g e b r a of an r - D i s c r e t e P r i n c i p a l Groupoid

Reduced C * - a l g e b r a s of r - d i s c r e t e p r i n c i p a l groupoids are g e n e r a l i z a t i o n s in a l l essential respects of the usual * - a l g e b r a s of matrices. They appear in a diagonalized form. That i s , C*(G O) is a maximal abelian subalgebra, the image of a unique condit i o n a l expectation. The elements of Cred(G,~ ) are matrices over G, the diagonal matrices are the elements of C*(G O) and the expectation map is e v a l u a t i o n on the

98

diagonal. The ideal s t r u c t u r e of Cred(G,~ ) is e a s i l y described. Ideals correspond to open i n v a r i a n t subsets of the u n i t space. Part of the representation theory may be conveniently expressed in terms of the groupoid. For example, the regular represent a t i o n on ~ is primary [resp. type I , I I or 111] i f f type I , I I or I I ~ .

the measure ~ is ergodic [resp.

Such C*-algebras are characterized by the existence of a special

kind of maximal abelian subalgebras, which, in accordance with [ 3 ~ , where a s i m i l a r notion is introduced in the context of von Neumann algebras, we c a l l Cartan subalgebras. In the f o l l o w i n g proposition, we use the reduced norm II fired, which has been defined in 2.8 and the sup-norm II II~. 4.1. Proposition : Let G be an r - d i s c r e t e groupoid with Haar system and l e t ~ be a continuous 2-cocycle. Then, the f o l l o w i n g i n e q u a l i t i e s hold f o r any f e Cc(G,~ ) : (i) (ii)

IlfII~ ~ IIfnred ; and f o r any u ~ GO,

Jlfl2d~u ~ Itftl~ed .

The proof r e s u l t s d i r e c t l y from the f o l l o w i n g lemma. Lemma : Let 8 and o be as above and l e t x be a point in G with d(x) = u. Consider the representation

L of Cc(G,o ) induced by the point mass at u (see 2.7). Let ~ and n

be the u n i t vectors au and ax r e s p e c t i v e l y in the space L2(G,~u) of the representation L. Then f o r any f ~ Cc(G,~), f ( x ) = ( L ( f ) ~ , n ) and f ( y ) = L ( f ) ~ ( y ) f o r any y c Gu. Proof : This is immediate since L is given by ( L ( f ) ~ , n ) = f f ( Y z ) ~ ( z - I ) ~(Y) ~(YZ,Z-1)d~U(z)d~u(y ) (see 2.7). Note also t h a t , since G is r - d i s c r e t e , ~u is the counting measure on Gu(see 1.2.7). For the proof of the proposition,

note that I f ( x ) l

~ IIL(f)I 1 II~II llnll < IIflIred by d e f i n i t i o n 2.8, and so

~ I f ( y ) I 2 d ~ u ( y ) = IIk(f)~II2 ~ IIL(f)II 2

II~II2 ~ IIfIIre dQ.E.D.

The i n j e c t i o n j of Cc(G ) into Co(G), the Banach space of continuous functions on G which vanish at i n f i n i t y , into Co(G) .

extends to a norm decreasing l i n e a r map j of Cred(G,~ )

99

4.2. Proposition

: Let G be an r - d i s c r e t e

groupoid with Haar system and ~ a continuous

2-cocycle. Then ( i ) the map j from C~ed(G,o ) to CO(G) is one-to-one ( t h e r e f o r e , of Cred(G,~) * will (ii)

the elements

be viewed as functions on G) ;

any a c C*red(G,o) s a t i s f i e s

IIaIL

_< llaIlred and II al 2 IIi

-< IIal1~ed

where the norm II III has been defined in 1.4 (llall I may be i n f i n i t e ) (iii)

_<

llall~,

; and

the operations in the * - a l g e b r a Cred(G,~ ) may be expressed in the same way as

in the same way as in the

. - a l g e b r a Cc(G,~), e x p l i c i t l y

a • (x) = a(x - I ) ~(x,x-1 ) a.b(x)

* , for a ~ Cred(G, ~,

= a(xy)b(y - I ) ~(xy,y-1)d~d(X)(y),

ha(x) = hor(x) a(x) Proof :

for a,b cCred(G,~ ), and , for h ~ C#GO) and a ~ C~ed(m,o ).

( i ) Let ~ be a q u a s i - i n v a r i a n t

probability

measure on GO. The regular represen-

t a t i o n on p is realized in standard form on L2(G,v -1) (see 1.10). This representation is the GNS representation associated with the state ~oP(f) = ~P(f)d~ where P is the r e s t r i c t i o n

map Cc(G,~ ) ÷ Cc(GO) and #0 is the c h a r a c t e r i s t i c

of GO, considered as a unit vector in L2(G,~-I). separating f o r the l e f t

= (L(f)~o,~o)

representation.

In p a r t i c u l a r ,

~0 is c y c l i c and

We may w r i t e L(f)6 0 = f . @0 = ~ ( f )

f ~ Cc(G,o ) where j is the natural map from Cc(G ) into

L2(G v - l ) .

a.e., j(a) = 0 :>~(a)

sentations form a f a i t h f u l (ii)

By c o n t i n u i t y ,

inequality (iii)

for

We note that by

~ 4.1 llj(f)ll ~ IIfIIred . Hence the e q u a l i t y remains true for a ~ C red(G,~)

As ~(a) = j ( a ) - 1

function

~

L(a)# 0 = _j ( a )

= 0 ~ > L(a) = O. Since the regular repre-

family of representations of C* red(G,a), a = O.

the i n e q u a l i t i e s

of 4.1 s t i l l

hold for a ~ C*red(G,o). The

Iiailre d ~ IIaNI has been w r i t t e n here for completeness. I t suffices to j u s t i f y

the passage to the l i m i t

in the expressions which are

valid for f ~ Cc(G,o ). For example, suppose that fn ÷ a and gn ÷ b in Cred(G,o ), with fn,gn ~ Cc(G,~ ). Then fn * gn (x) = ~ fn(xy)gn ( y - 1 ) ~ ( x y ' y - 1 ) d ~ d ( x ) ( y ) " estimate II IL

~ II Ilre d, fn * gn (x) ÷ a * b ( x )

(a*b

denotes the product of a and b).

On the other hand, because of the estimate II II2 ~ II fire d where L2(G,~d(x)),

fn(X.)

÷a(x.)

and gn ÷ b in L2(G,~d(x)),

Because of the

II II2 is the norm of

hence the r i g h t hand side goes

•

100

to rj a ( x y ) b ( y - 1 ) ~ ( x y , y - 1 ) d x d ( X ) ( y ) . Q.E.D. Remark :

(Cf.

[31,1Z]).

case of the p r i n c i p a l

I t seems hard to c h a r a c t e r i z e the range of the map j .

In the

groupoid I x I , where I is a countable d i s c r e t e space, t h i s

amounts to c h a r a c t e r i z i n g the matrices of compact operatorS. We may note t h a t , in t h i s case, the c o n d i t i o n s

S l a ( x ) I 2 d~U(x) <_ IIall 2 are s a t i s f i e d by the matrix o f any

bounded operator. Let us study now the ideal s t r u c t u r e of the reduced C*-algebra of an r - d i s c r e t e principal

groupoid. In f a c t one can do a l i t t l e

4.3. D e f i n i t i o n

:

better.

Let us say t h a t a l o c a l l y compact groupoid G is e s s e n t i a l l y p r i n c i -

pal when f o r every i n v a r i a n t closed subset F of i t s u n i t space, the set of u's in F whose i s o t r o p y group G(u) is reduced to {u} is dense in F. 4.4. Proposition :

Let G be an r - d i s c r e t e e s s e n t i a l l y p r i n c i p a l groupoid with Haar

system. Then f o r any q u a s i - i n v a r i a n t measure ~, any ~ - r e p r e s e n t a t i o n L of G on and any f c Cc(G), the f o l l o w i n g i n e q u a l i t y holds : sup ucF

If(u)l


where F is the support of u.

Proof : I t s u f f i c e s to prove the i n e q u a l i t y

If(y)I<

llL(f)I I

f o r y E F such t h a t G(u) = { y } .

Let (Vn) be a fundamental sequence of neighborhoods of u. There e x i s t s a sequence (~n) of s q u a r e - i n t e g r a b l e sections of the H i l b e r t bundle of the representation s a t i s f y i n g Supp ~nC Vn We w i l l

and f ] ~ n ( U ) l 2 d~(u) = I

show t h a t the sequence ( L ( f ) ( n ,

(n) tends to f ( u ) .

We f i r s t

w r i t e f as a f i n i t e sum o f f u n c t i o n s supported on compact open G-sets : m f = ~ fi " f i = h. ×Si with h i a Cc(GO) and Si c 9. 1 We use 1.7. ( * ) to compute ( L ( f i ) ~ n , (n) : (L(fi)(n'(n)

= SVnnV n

S#1

hi(u)(L(u Si)~n(U.Si),~n(U))

D-I/2(u Si) d#(u)

I f y ~ y -S i , we have e v e n t u a l l y Vn n v n - S i I = ~ and ( L ( f i ) ~ n, ~n ) = 0 .

I01 I f u = u . Si , the G-set Si meets GO. For n large enough, VnSi is contained in GO and (k(fi)~n,~n)

= ~Vn h i ( u ) ll~n(U)ll2d~(u)

tends to h i ( u ). Q.E.D.

Let

G be an a r b i t r a r y

l o c a l l y compact groupoid with Haar system. I t s reduced

C -algebra has a d i s t i n g u i s h e d f a m i l y of i d e a l s , defined by i n v a r i a n t open subsets of GO. This is well known in the cases studied p r e v i o u s l y (e.g. introduce some n o t a t i o n . O(G) w i l l

J(A) w i l l

denote the l a t t i c e

denote the l a t t i c e

[86], 2.29). Let us

o f ideals of the C * - a l g e b r a A.

of i n v a r i a n t open subsets of the u n i t space o f the

groupoid G. For U in O(G), Ic(U ) = { f ~ Cc(G,~ ) : f ( x ) = 0 i f x ~ GU } and I(U) is the closure of Ic(U ) in C* (G,o) red Lemma :

Let X be a l o c a l l y compact space and Y be a normal open subspace. Then,

the closure of {f ~

{ f c Cc(X ) : s u p p f c Y } in the i n d u c t i v e l i m i t

Cc(X ) : f ( x ) = 0 i f x ~ Y}.

Proof :

One constructs an approximate i d e n t i t y

increasing nets ( V )

and (V',~ ) of r e l a t i v e l y

f o r Cc(Y ) as f o l l o w s . "There are

compact open subsets of Y such t h a t

~ c V'm w h i l e uV~ = uV'm = Y and there are f u n c t i o n s e V'm

topology of Cc(X) is

such t h a t em = 1 on Vm

and f e

If

÷ f in the i n d u c t i v e l i m i t

c

Cc(Y ) supported on

f c Cc(X ) and f ( x ) = 0 i f x # Y~ supp ( f e ) c Y topology, Moreover, the set { f E C (X) : f ( x ) = 0

i f x # Y} is c l e a r l y closed. 4.5. Proposition :

Let G be a l o c a l l y compact groupoid w i t h Haar system and l e t

be a continuous 2 cocycle. (i)

If

U is an i n v a r i a n t open subset of GO and F is i t s complement, then I(U)

is an ideal of C* (G,a) which is isomorphic to C~ed(Gu,~) and such t h a t the q u o t i e n t red is isomorphic to Cred(GF,~ ). (ii)

I f u be a q u a s i - i n v a r i a n t measure of support F, the ideal

I ( U ) , where

the complement of F, is the kernel of the r e g u l a r representation on ~. (iii)

The correspondence U ~ I(U) is a one-to-one order preserving map from

O(G) i n t o ~(Cred(G,~)).

U is

102 Proof : ( i ) Using the i n v a r i a n c e of

U, one e a s i l y checks t h a t I c ( U )

is a s e l f - a d j o i n t

two-sided ideal o f Cc(G,a ). Indeed, suppose f c Ic(U) and g c Cc(G,a), then f * g (x) = J f ( y ) g ( y - L x ) a ( y , y - l x )

d~r(X)(y).

I f x # GU and r ( y ) = r ( x ) , y # GU and f ( y ) = O, hence f , g(x) = O. Therefore i t s closure in Cred(G,a ) is a closed ideal of C;ed(G,~ ). The map j from Cc(Gu,~ ) to Cc(G,a ) which extends a f u n c t i o n on GU by 0 outside o f GU is a *-homomorphism and is i s o m e t r i c f o r the reduced norm. In f a c t ,

i f we compose i t with the r e g u l a r

r e p r e s e n t a t i o n on ~, where u is a q u a s i - i n v a r i a n t measure on G, we o b t a i n the r e g u l a r r e p r e s e n t a t i o n on UU' the r e s t r i c t i o n i n v a r i a n t measure on Indp

of u to U. Conversely, i f ~ is a quasi-

U, i t can be viewed as a q u a s i - i n v a r i a n t measure ~ on GO and

( f ) = Ind~ ( j ( f ) )

f o r f c Cc(GU,~ ). Hence we have an isometric

*-homomor-

phismfrom Cred(Gu, a) to Cred(G,a ). The lemma shows t h a t i t s image is I(U). The r e s t r i c t i o n

map p from Cc(G,a ) onto Cc(GF,~) is a *-homomorphism. I f u is

a q u a s i - i n v a r i a n t measure on F, we view i t as a q u a s i - i n v a r i a n t measure on GO, say 2. We have

Indu ( p ( f ) )

= Ind~ ( f ) f o r f c Cc(G,c ). Hence p decreases the reduced norm and

extends to a*-homomorphism from C* red(G,~) onto C;ed(GF,a)

I t s kernel I c l e a r l y con-

t a i n s I(U). Let L be a r e p r e s e n t a t i o n o f Cred(G,a ) which vanishes on I(U). Ne d e f i n e LF on Cc(GF,~ ) by LF(f ) = L ( f ' )

where f '

~ Cc(G,a ) and f ' i G F = f . This makes sense

because L vanishes on Ic(U ). The map LF is a r e p r e s e n t a t i o n of Cc(GF,{ ) and s a t i s f i e s LFOp (f) = L ( f ) . Indu 2 are d i s j o i n t

I f u I and ~2

are d i s j o i n t q u a s i - i n v a r i a n t measures on GO, Indu I and

r e p r e s e n t a t i o n s and Illnd~mV~2(f')H =

This gives the estimate

HLF(f)IJ<]JLII ljfl]red.

Max(IIIndum(f')ll, l l I n d u 2 ( f ' ) H ) .

Hence LF extends to a r e p r e s e n t a t i o n of

Cred(GF,a ) and L f a c t o r s through p. Therefore I - I(U). (ii)

I t s u f f i c e s to show t h a t the r e g u l a r r e p r e s e n t a t i o n on a q u a s i - i n v a r i a n t

measure u of support GO is a f a i t h f u l

r e p r e s e n t a t i o n of C;ed(G,a ). Let M be a represen-

t a t i o H of GO. I t is weakly contained in the r e p r e s e n t a t i o n defined by ~. Since the process o f i n d u c t i o n preserves weak containment, the kernel of IndM contains the kernel of Indu. (iii)

This is c l e a r . Q.E.D.

103 We are ready to give the announced r e s u l t on the ideal s t r u c t u r e of the reduced C * - a l g e b r a of a p r i n c i p a l

r - d i s c r e t e groupoid. I t is well known in the case of a

transformation group ( c o r o l l a r y 5.16 of [24] and theorem 5.15 of [86]). l,lithout the assumption of r - d i s c r e t e n e s s ,

the problem has r e c e n t l y been solved, in the case of a

transformation group, by E.Gootman andJ.Rosenberg [38]. 4.6. Proposition

:

Let G be an r - d i s c r e t e e s s e n t i a l l y p r i n c i p a l groupoid w i t h Haar

system and a a continous 2-cocycle. Then the correspondence U ~ I ( U ) is an order preserving b i j e c t i o n

between the l a t t i c e Lg(G) of i n v a r i a n t open subsets of GO and the

J ( rCe*d ( G , a ) )

lattice Proof :

Let L be a

of i d e a l s o f the reduced C*-algebra Cred(G,a) * o - r e p r e s e n t a t i o n of G l i ving on the q u a s i - i n v a r i a n t p r o b a b i l i t y

measure ~. Let P be the r e s t r i c t i o n

map from Cc(G,~ ) onto Cc(GO). We have seen

(comments before 2.7) t h a t P is a c o n d i t i o n a l expectation and t h a t Indu is the GNS representation associated w i t h the state uoP. I t r e s u l t s from 4.4 t h a t luoP(f) l ~ IIm(f) lI, hence jIlnd~ ( f ) I I
f o r any f

c

Cc(G ).

i f L is a representation of Cred(G,~ ), i t s kernel is contained in I(U)

where U is the complement of the support of ~. Since the reverse i n c l u s i o n is c l e a r , i t s kernel is p r e c i s e l y I ( U ) . Hence the map U ~ I(U) is onto. Our next task is to j u s t i f y r-discrete 4.7.

principal

Q.E.D. the statement t h a t the reduced C~ - a l g e b r a of an

groupoid appears in a diagonalized form.

Proposition : Let G be an r - d i s c r e t e groupoid w i t h Haar system and a a c o n t i -

nuous 2-cocycle. Then (i)

an element a of Cred(G,a ) commutes w i t h every element of C*(G O) i f f

vanishes o f f the i s o t r o p y group bundle G' = {x E G : d(x) = r ( x ) } (ii) Proof :

C* (GO) is a maximal subalgebra of C~ed(G,a) i f f

it

; and

GO is the i n t e r i o r

Since GO is open, Cc(GO)is a subalgebra of Cc(G,~) and

of G'.

C~(GO) is a

subalgebra of Cred(G,a ). I t consists e x a c t l y of those elements o f Cred(G,a) which vanish o f f the u n i t space GO. Let a E C~ed(G,a) and h ~ C* (GO). Then ah(x) = a(x)hod(x) and ha(x) = hor(x) a ( x ) .

I f a(x) = 0 f o r any x such t h a t d(x) # r ( x ) ,

then a(x)hod(x) = h o r ( x ) a ( x ) holds f o r every x i n G. I f a(x) # 0 f o r some x such t h a t d(x) # r ( x ) ,

then there e x i s t s h ~ C*(G O) such t h a t hod(x) = I and hor(x) = O,

consequently, a(x)hod(x) # h o r ( x ) a ( x ) , and so ah # ha. The assertion ( i i ) immediate consequence of ( i ) .

is an Q.E.D.

104 4.8. P r o p o s i t i o n :

Let G be an r - d i s c r e t e p r i n c i p a l groupoid w i t h Haar system and map P : C:ed(G,a) --~ C~( m0 ) is the

a continuous 2-cocycle. Then the r e s t r i c t i o n

unique c o n d i t i o n a l expectation onto C*(G O) and is f a i t h f u l . Proof :

The proof is i d e n t i c a l to a proof one would give in the case of m a t r i x

algebras. Note t h a t , by 4.2, P is well defined. There is no d i f f i c u l t y P has a l l

checking

the p r o p e r t i e s of an e x p e c t a t i o n map. To show uniqueness, we use the same

device as in 4.3 or 4 . 5 . d . Let a ~ Cc(G,~ ) and suppose t h a t suppa does not meet the diagonal A of finite

GO × GO - again, we view G as a subset of GO × GQ. There e x i s t s a

cover of r(supp a) by open sets Ui i=1 . . . . . n on GO such t h a t Ui × Ui n supp a

= @ f o r i=1 . . . . . n. Let ( h i } be a p a r t i t i o n of u n i t y subordinate to t h i s cover, with n ~ h11/2a ~hi(u ) = 1 f o r u ~ r(supp a). Then a = ( ~ hi) a and 0 = • hi1/2. l l c o n d i t i o n a l e x p e c t a t i o n onto C ~(GO), then n n o=

Q( hil/2ahi Ij2) = i hil/2QIa)hi112=

I f Q is any

h QIa)

n = Q( # hia ) = Q(a) . Since & is closed and open in G, an a r b i t r a r y a in Cc(G,~ ) may be w r i t t e n a = a I + a2 where a I is the r e s t r i c t i o n

of a to & and s u ~ a 2 does not meet A. Consequently, Q(a)

= a 1. This shows t h a t Q agrees with P on Cc(G,~ ), hence on Cred(G,~ ). To see t h a t P is f a i t h f u l ,

note t h a t i f a c Cred(G,~), then P(a ~ * a)(u) = J i a ( x - 1 ) I 2 d~U(x).

Hence i f P ( a * ~ a) = O, then a(x) = 0 f o r a l l x. Q.E.D.

4.9. D e f i n i t i o n :

Let A be a C * - a l g e b r a and B an abelian sub C~ - a l g e b r a . We c a l l

normalizer of B ( i n A) the inverse semi-group ~kC(B) = { a , p a r t i a l

isometry of A : d ( a ) , r ( a )

and r ( a ) denote the i n i t i a l an isomorphism sa : b ÷ aba~

~ B and a(Bd(a))a * = Br(a)}

where d(a)

and f i n a l p r o j e c t i o n s of a. An element a m J~r(B) induces of Bd(a) onto Br(a) ; we also denote the corresponding

p a r t i a l homeomorphism o f d(a) onto r ( a ) in the spectrum B o f B by s a. The inverse semi-group of p a r t i a l homeomorphisms of B of the form s a with a c A r ( B )

is c a l l e d the

ample semi-group of (B,A) (or of B when there is no ambiguity) and is denoted ~ ( B ) .

105 The semi-group of p a r t i a l isometries of B is denoted%(B).The u n i t spaces ofJ~r(B), ~(B) and ~(B) can a l l be i d e n t i f i e d with the Boolean algebra

~of

p r o j e c t i o n s of B.

Remark : I f B is a maximal a b e l i a n subalgebra o f a C * - a l g e b r a A,

~%(B) ÷~°(B)

÷ g

(B) ~

is an exact sequence o f inverse semi-groups (see 1.1.17). Indeed, i f s a is an potent in

idem-

~ ( B ) , then ab = ba f o r any b c B, hence a c JC(B) n B =cU~(B).

Recall t h a t the ample semi-group of an r - d i s c r e t e groupoid is the semi-group o f i t s compact-open G-sets. In the case o f a p r i n c i p a l groupoid, G-sets are uniquely determined by the p a r t i a l transformations they induce on the u n i t space. Therefore can be viewed as a semi-group of p a r t i a l homeomorphisms of the u n i t space o f the p r i n c i p a l groupoid. 4.10. P r o p o s i t i o n : Let G be an r - d i s c r e t e p r i n c i p a l groupoid with Haar system and l e t ~ be a continuous 2-cocycle. Then the ample semi-group ~(B) of the maximal a b e l i a n subalgebra B = C * ( G O) of the C * - a l g e b r a A = C* (G,a) coincides with the ample red semi-group ~ o f the groupoid. Proof :

We f i r s t

show t h a t ~ i s

contained in ~(B). I f s is a compact-open G-set,

i t s c h a r a c t e r i s t i c f u n c t i o n Xs is a p a r t i a l as an obvious computation shows : ×s * ×Z

isometry in Cc(G,o ) which normalizes B, = r ( s ) , where r ( s )

c h a r a c t e r i s t i c f u n c t i o n , Xs . ×s = d ( s ) , and ×s* h , × s* hS(u) = h(u-s) i f u ~ r ( s ) and 0 i f u # d(s).

is i d e n t i f i e d with i t s

= hs f o r h c Bd(s), where

Hence Xs induces the p a r t i a l homeo-

morphism u ~ u.s. Conversely, suppose t h a t a is in J~C(B) and l e t s = s a the p a r t i a l homeomorphism i t

i nduces on GO. We want to show t h a t i t s graph is a compact open G-set.

As in p r o p o s i t i o n 2.9 o f [31, I l l , a . h .

a simple computation shows t h a t

a * ( u ) = I [ a ( Y ) [ 2 hod(y)d~U(y)

f o r h ~ Cc(d(s)). By d e f i n i t i o n ,

t h i s equals h(u.s) f o r u e r(s).We f i x u e r ( s ) .

I f y ~ G~(s) does not belong to s, then d(y) # u.s and there e x i s t s an non-negative f u n c t i o n h e Cc(d (s)) such t h a t hod(y) = l a n d h(u.s) = 0 ; t h i s implies a(y) = O. Hence a(y) = 0 i f y ~ s. Moreover, l a ( y ) I

= 1 i f y belongs to s. Since a is in

Cc(G), s must be a compact open set of G. Q.E.D.

106

A l a s t property of the pair ( C r e~d ( G , ~ ) , C*(GO)) needs to be interpreted in terms of the groupoid. This is the notion of regular abelian subalgebra introduced in Dixmier ~7] in the context of von Neumann algebras. 4.11. D e f i n i t i o n : An abelian s u b - * - a l g e b r a B of a C * - a l g e b r a A is said to be regular i f the l i n e a r span of the elements of the form ab, where a ~ J~r(B) and b ~ B, is dense in A. 4.12. Proposition :

Let G be an r - d i s c r e t e groupoid with Haar system and a a

red(G,~) i f f continuous 2-cocycle. Then C*(G O) is a regular subalgebra of C*

G can be

covered with compact open G-sets. Proof :

I f G can be covered with compact G-sets, then one can, by using a p a r t i t i o n

of the u n i t y , w r i t e any f c Cc(G,~) as a sum of functions supported on compact open G-sets and a function supported on the compact open G-set s may be w r i t t e n under the form X s .

h where h c Cc(GO). Conversely, i f the space of continuous functions

supported on compact open G-sets is dense in C r*e d (G, ~ ) , they cannot a l l vanish at a given point x of G, Consequently such point x is contained in a compact open G-set. Q.E.D. The properties of the subalgebra C*(GO), when G is an r - d i s c r e t e principal groupoid, may be summarized by introducing, as in d e f i n i t i o n 3.1 of ~ I ,

I~,

the

notion of Cartan subalgebra. Recall that the ample semi-group of an r - d i s c r e t e p r i n c i pal groupoid with Haar system has the property of acting r e l a t i v e l y f r e e l y on the u n i t space (1.2.14), in the sense that the set of fixed points of each of i t s elements is open. 4.13. D e f i n i t i o n : An abelian sub-*-algebra B of a C * - a l g e b r a w i l l be called a Cartan subalgebra i f i t has the f o l l o w i n g properties : (i) (ii) (iii) (iv)

i t is maximal abelian ; i t is regular ; i t s ample semi-group~(B) acts r e l a t i v e l y f r e e l y on i t s spectrum B ; and the exact sequenceS~÷°d~(B) ÷ ~ ( B ) ~ ~(B) ÷SSsplits in the sense that there

exists a section k for s s a t i s f y i n g k(se) = k(s)e, k(es) = ek(s) and k(e) = e, for every e in

S~and s in ~ ( B ) .

107 Question :

Is ( i v ) independent of ( i ) - ( i i i )

4.14. Proposition :

?

Let G be an r - d i s c r e t e p r i n c i p a l groupoid admitting a cover of

compact open G-sets and l e t a be a continuous 2-cocycle. Then C* (GO)is a Cartan subalgebra of C * (G,a). red Proof : I t is maximal abelian by 4 . 7 . ( i i ) ,

regular by 4.12 and i t s ample semi-group

~ , which is the ample semi-group of G by 4.10, acts r e l a t i v e l y f r e e l y on GO by 1.2.13. A section for J~r(C*(GO)) + ~ i s

given by k(s) = Xs where

Xs is the

c h a r a c t e r i s t i c function of the G-set s. Q.E.D. This proposition admits a converse. 4.15. Theorem :

Let B be a Cartan subalgebra of a separable C*-algebra A.

( i ) There exists an r - d i s c r e t e principal groupoid G admitting a cover by compact open G-sets, a continuous 2-cocycle ~ and a *-homomorphism ~ of C* (G,~) onto A which carries f a i t h f u l l y

C*(G O) onto B and the ample semi-group of G onto the ample semi-

group of B. (ii)

The groupoid G is unique up to isomorphism and the 2-cocycle a is unique up

to a coboundary. (iii)

I f G is amenable, the*-homomorphism ~ is an isomorphism and B is the image

of a unique conditional expectation, which is f a i t h f u l . Proof : (i)

The ample semi-group ~(B) of B is an inverse semi-group of p a r t i a l homeomor-

phisms of B, defined on compact open sets. By 4.13. ( i i i )

and 1.2.13, the principal

groupoid G associated to i t has a structure of r - d i s c r e t e groupoid with Haar system such that B becomes i t s u n i t space and ~(g) i t s ample semi-group. Let k be a section for s as in 4.13 ( i v ) . By d e f i n i t i o n of s, i t s a t i s f i e s the covariance property k ( t ) a k(t)

*

=

at

f o r each t c ~(B) and each a eqJ~(B). Hence, the extension is compatible

with the action of ~(B) on qd~(B) (see 1.1.17). As in 1.1.17, t h i s extension is d e f i ned by a 2-cocycle } E Z 2 ( g ( B ) , ~ L ( B ) ) . As in 1.2.14, there exists a unique continuous

108 2-cocycle a s Z2(G,T) such that a(s,t)(u)

= a(us,ust)

for every s , t s ~ ( B ) . n

We t r y to define a map # of Cc(G,a ) into A by the formula

n

¢(~ h i Xsi ) = # h i k ( s i ) where h i s Cc(GO) c C * ( G O) = B and s i s ~ ( B ) . The map ¢ is n

well defined.First,n that

any element of Cc(G,a)may be written ~l hi Xsi" Second, suppose

~ h i ×si = 0 : find d i s j o i n t compact open G-sets t j ,

j = 1,...,m such that each

s i may be expressed as an union of t j ' s .

We may write s i = .u s i j t j where s i j = 0 or I J = t j - Then k(si) = JZ. s i j k ( t j) and ×s i = ~J s i j × t j . The equality

and Otj = 0 , l t j

Z.(~ ~ijhi ) x t j = !. hi×si = 0 implies j 1 are d i s j o i n t ,

!. s i j h i l r ( t j

l herefore Z hik(s i) = Z.(~. s i j h i ) k ( t j ) = O. The same argument shows that 1

j l

¢ is one-to-one. We note that ¢ is a*-homomorphism. ×~

= ~(s-ls)

~(×~ )

) = 0 for each j , because the t .J' s

For i f s is a G-set, then

* ×s-l,

: ~(s-i,s) * k(s -i) = [k(s -1) k(s) k ( s - l s ) *] ~ k(s -I) = k(s)*

= ¢(XS)* ;

and i f s and t are G-sets, then Xs . Xt = ~(s,t) ×st ¢(Xs . ×t) = ~(s,t) k(st)

= k(s) k(t) k ( s t ) * k ( s t )

= k(s) k ( t )

: @(×s) ¢(×t ). The map ¢ is continuous when Cc(G,~) has the inductive l i m i t topology. Indeed, l e t ( f i ) converge to f in the inductive l i m i t topology ; multiplying by a f i n i t e partition of unity, we may assume that the f i ' s and f have their support contained in a common compact open G-sets ; the assertion is now obvious. We may apply theorem 1.21 (or rather, i t s corollary 1.22) to conclude that ¢ i s continuous for the C -norm of Cc(G,a). Since A is separable ; B is separable and ~(B) is countable, hence G is second countable. We have already noted (1.3.28) that a second countable r-discrete groupoid has s u f f i c i e n t l y many non-singular G-sets. Thus ¢ extends to a *-homomorphism of C (G,a) into A. I t is onto because i t s range contains the elements ab with

109 a

~ ~(B) and b e B and B is r e g u l a r . (ii)

The groupoid G is uniquely defined by B and i t s ample semi-group. The 2-cocy-

cle ~ is determined up to a coboundary by the extension

~÷%(B) o.~(8)+ (iii)

~ (8) ÷ ~ .

I f G is amenable, proposition 3.2 shows that C*(G,~) = Cr~d(G,~ ).

The kernel of ¢ is an ideal of Cred(G,o ) which intersects Cc(G,o ) t r i v i a l l y , is t r i v i a l

hence i t

by 4.6. The l a s t assertion results from 4.8. Q.E.D.

4.16. Remark :

I t w i l l be given in 3.2.5 an example of a C * - a l g e b r a with two

maximal abelian sub C * - a l g e b r a s , one of which is a Cartan subalgebra, the other satisfies (i)(ii)

but not ( i i i )

of 4.13 and is the image of a unique f a i t h f u l condi-

tional expectation. Let us conclude t h i s section by r e c a l l i n g some f a c t s , due to P. Hahn, pertinent to the regular representations of a p r i n c i p a l groupoid. The case of an r - d i s c r e t e p r i n c i p a l groupoid is studied in [ 3 ~ . 4.17. Proposition : (P.Hahn [ 4 ~ ) .

Let G be a second countable l o c a l l y compact groupoid

with Haar system, ~ a continuous 2-cocycle and u a q u a s i - i n v a r i a n t measure on GO. Then ( i ) the a-regular representation on ~

(defined in 1.8) is a factor representa-

t i o n i f f v is ergodic, (ii) Ill)

i t is of type I (resp. I I I ,

II , Ill)

iff

~ is of type I (resp. I I 1, I I ,

(defined in 1.3.13).

Proof :

The assertion ( i ) and part of the assertion ( i i )

r e s u l t from his theorem ~.1.

The rest results from his theorems 5.4 and 5.5.

5. Automorphisms Groups, KMS States and Crossed Products

This section i l l u s t r a t e s the use of groupoids in the study of basic

problems

110 f o r C * - a l g e b r a s . As usual G denotes a l o c a l l y compact groupoid with Haar system {~u} and ~ a continuous 2-cocycle. Here A denotes a l o c a l l y compact abelian group with dual group A. The value of the character ~ ~ A at a ~ A is w r i t t e n (~,a). Let c ~ ZI(G,A) be a continuous one-cocycle. Define, f o r each ~ ~ A, ~(f)(x)

= (~,c(x))f(x)

5.1. P r o p o s i t i o n : (i

~

f o r f c Cc(G,o ).

Let c ~ ZI(G,A) and ~ be as above. Then

is an automorphism of Cc(G,~ ) ;

mg extends to an automorphism o f C~ (G,{)

(ii

;

(C * (G,~),A,m) is a C* -dynamical system (see 7.4.1 in [ 6 0 ] ) , in other

(iii words

is a continuous homomorphism of A i n t o the group Aut(C ~ (G,o)) o f automorphisms

of C

(G,~) equipped with the topology of pointwise convergence ; and (iv)

~ leaves C* (GO) pointwise f i x e d .

Proof : (i) (ii)

This is a r o u t i n e v e r i f i c a t i o n . F i r s t , one notes t h a t ~

f l~(f)I(x)d~U(x) Hence

is isometric with respect to the

111norm

-X-

~-1

P

I t is c l e a r t h a t m : A÷Aut(C (~,~)) is a group homomorphism. Let us check

its continuity.

I t s u f f i c e s to check t h a t the map g -~ mgf is continuous f o r any

f c Cc(G,o ) and f o r the topology of the It iTI norm. Let K be the support of f . > O, there e x i s t s a neighborhood V o f ~ in A such t h a t f o r n c V I ( n , c ( x ) ) < c

:

= I If(x)Id~U(x).

m~ is continuous with respect to the C* -norm and so is ~ - I =

(iii)

I[

f o r any x c K. Then ]l~nf - ~ f l 1 1 , r

sup l~nf - ~g

For any - (~,c(x)) 1

d~u _<

U

Hence, II~ f (iv)

~fll I

_< ~Ilfll I .

Clear. Q.E.D.

In the case when A is the group of real numbers, one may define a l i n e a r map on the domain D(~) = Cc(G,{ ) by (~f)(x) = ic(x)f(x) 5.2. P r o p o s i t i o n . Let c e Z I ( G , ~ ) ,

and l e t m and 6 be as above. Then

111 (i)

Cc(G,a ) consists of e n t i r e a n a l y t i c elements f o r m ; and

(ii)

~ is a * - d e r i v a t i o n

d e r i v a t i o n s on C * - a l g e b r a s Proof :

and a pregenerator f o r

~. (A reference f o r unbounded

is [ 6 5 ] ) .

We note t h a t f o r any continuous f u n c t i o n # on G and f ~ Cc(G,a ),

!l~ftl ~ ll~flli s (sup !¢(x)l)lfftli x~K

where #f denotes the pointwise product and K is the support of f . Therefore, Cc(G,a ) is in the domain of the generator of m and ~ is i t s r e s t r i c t i o n , atf - f I}--~-

6f II ~ sup l K

eitc(x) t

- 1

ic(x)]

and

IIfl] I .

The same argument shows t h a t 6nf e x i s t s f o r any i n t e g e r n and II~nfll

~ (sup K

I c ( x ) i ) n llflIl -

This proves the f i r s t

a s s e r t i o n . Also, 6 is closable and i t s closure generates an

automorphism group which can be nothing but generator of

~. Hence, the closure of a is the

m. Q.E.D.

Let us say t h a t an automorphism group m~ of a C * - a l g e b r a e x i s t s a group of u n i t a r i e s U~ (i) (ii)

m~(A) = U~A U~~

i n the m u l t i p l i e r

topology on the m u l t i p l i e r f o r B in the o r i g i n a l 5.3. Proposition :

algebra such t h a t

f o r any element A of the C * - a l g e b r a ,

~ ÷ U~ is continuous f o r the s t r i c t

is inner i f there

topology.

and

(Recall t h a t the s t r i c t

algebra is defined by the semi-norms A ÷ IIABII and A ~ IIBAII

algebra).

Let c ~ ZI(G,A) and l e t ~ be the associated automorphism group

of C * ( ~ , a ) . (i) (ii) Proof

I f c c BI(G,A), then m is i n n e r . I f G is r - d i s c r e t e ,

principal

and amenable, the converse holds.

:

(i)

One f i r s t

observes t h a t any bounded continuous f u n c t i o n on GO defines in

1t2 the obvious way an element of the m u l t i p l i e r

algebra of C * ( G , ~ ) .

I f c(x) = bor(x) -

hod(x) where b is a continuous f u n c t i o n on GO, then f o r each ~ in A, U~(u) = ( ~ , b ( u ) ) defines a u n i t a r y element of the m u l t i p l i e r

algebra and f o r f e Cc(G,{),

~E(f)(x) = (~,c(x))f(x) = (U~f U*~ ) ( x ) . The c o n t i n u i t y of ~ ~ U~ is checked as in 5.1 ( i i i ) . (ii)

I f m~(A) = U~A U~

hence is i t s e l f

, then U~ commutes with every element of C* (GO) and

diagonal (see section 4 - we have not considered the m u l t i p l i e r

algebra t h e r e , but i t s elements can also be viewed as continuous functions on G). Therefore U~ is o f the form U~(u) = (E,b(u)) where b is a continuous f u n c t i o n on GO and c(x) = bor(x) - bod(x). Q.E.D. As an example, l e t us i n t e r p r e t the theorem 4.8 of the f i r s t assume t h a t G is an r - d i s c r e t e ,

chapter. We

p r i n c i p a l and amenable groupoid with compact u n i t

space. By 4.6 of t h i s chapter, C * ( G , ~ )

is simple i f f

G is minimal. Let c ~ zl(G,R)

and assume t h a t c is bounded. This amounts to saying the associated d e r i v a t i o n ~ is bounded, or e q u i v a l e n t l y , t h a t the associated automorphism group is norm continuous. Then the range of c

R(c) is compact and the asymptotic range R (c) is zero. The

theorem states t h a t i f G is minimal and c bounded, then c is in BI(G,A). In o t h e r words, i f C * ( G , ~ )

is simple and ~ bounded, then a is inner. This is a p a r t i c u l a r case

o f a well known r e s u l t o f Sakai ( [ 6 4 ] , 4 . 1 . 1 1 ) . When G is r - d i s c r e t e ,

p r i n c i p a l and amenable, the asymptotic range R (c) of a

cocycle c ~ ZI(G,A) can be i d e n t i f i e d as the Connes spectrum (see [60], 8.8.2) of the associated automorphism group. This is h e u r i s t i c a l l y definitions

c l e a r when one compares both

:

R(c) = nR(cu) where U runs over a l l non-empty open sets in GO, c U is the r e s t r i c t i o n

of c to GIU,

and R(c) is the closure o f c(G) ; w h i l e r(~) = nSp(~lB ) where B runs over a l l

~ - i n v a r l a n t , h e r e d i t a r y non-zero sub C * - a l g e b r a s of C* (G,~),

and Sp(m) is the Arveson spectrum of m([60] 8 . 1 . 6 . ) .

I t can be seen in our case t h a t

113

Sp(m) = R(c) and t h a t f o r every open set U in GO, C * ( G I U , ~ ) may be viewed as an m - i n v a r i a n t h e r e d i t a r y subalgebra o f C * ( G , ~ ) .

(This may be done in a fashion

analogous to 4 . 4 ) . However, in order to avoid the e x p l i c i t h e r e d i t a r y subalgebras o f C * ( G , o ) , we w i l l

d e t e r m i n a t i o n o f the

use an a l t e r n a t e d e f i n i t i o n of r ( ~ )

8.11.8) which uses only the i d e a l s o f the cross-product algebra C ~(G,~) x A. will

([60], This

be done in 5.8. Given a cocycle c in Z I ( G , ~ )

and 5 s ~ , + ~ ] , t h e (c,~)KMS

measure u on GO has been defined in 1.3.15. I t is time to j u s t i f y

condition for a t h i s terminology.

We have seen how a one-parameter automorphism group m of C*(G,~) is associated to c. On the other hand, composing ~ with the r e s t r i c t i o n

map from Cc(G,~)onto Cc(GO),

one obtains a p o s i t i v e l i n e a r f u n c t i o n a l # = ¢~ on Cc(G,~ ). A p o s i t i v e l i n e a r funct i o n a l on Cc(G,~)continuous f o r the i n d u c t i v e l i m i t " p o s i t i v e type measure" - w i l l

top~)logy - an e q u i v a l e n t term is

be c a l l e d here a weight on C (G,~). This does not agree

with the usual d e f i n i t i o n of a weight on a C~ - a l g e b r a (see [12], page 61), Cc(G,o ) is not always a h e r e d i t a r y subalgebra o f C~(G,o), but i t

because

is convenient here.

I f G is r - d i s c r e t e and u a p r o b a b i l i t y measure, ¢ is a state. We note t h a t , with above n o t a t i o n s , ¢ is m - i n v a r i a n t since c vanishes on GO. 5.4. P r o p o s i t i o n : Let c c Z I ( G , R ) ,

B c [0, + ~] and ~ be a measure on G0. The

automorphism group associated with c is denoted by ~ and the weight associated w i t h is denoted by 0- Then the f o l l o w i n g p r o p e r t i e s are e q u i v a l e n t : (i

the weight ¢ s a t i s f i e s the (m,6)KMS c o n d i t i o n (see [60] 8.12.2 o r [ 6 ~ 6.1) ;

and (ii)

the measure ~ s a t i s f i e s the (c,B)KMS

is p r i n c i p a l and ~ f i n i t e ,

c o n d i t i o n ( 1 . 3 . 1 5 ) . Moreover, i f G

any weight ¢ which s a t i s f i e s the (~,~)KMS

c o n d i t i o n arises

from a measure u on GO. Proof :

We f i r s t

consider the case when # is f i n i t e .

Replacing c by 6c, we may assume

t h a t 6 = 1. (i) ~> any t c ~

(ii)

Since # is 1 - KMS f o r m, we f i n d t h a t f o r any f , g c Cc(G,g ) and

114

~[~t(~)*g]

= ~[g.~t+i(f~,

because f is a n a l y t i c f o r

~(5.2.(i)).

Let us evaluate both expressions.

For the f i r s t , #[at(f)~g]

= I eitc(y)

f(Y) g ( y - l ) d v ( Y ) ,

where v = I ~Ud~(u)'

w h i l e f o r the second, @[g * m t + i ( f ) ]

= Ig(Y) ei(t+i)c(y-1) = ~eitc(y)

In p a r t i c u l a r ,

f(y-1)

dr(y)

f ( y ) g(y-1) e-C(y) d - 1 ( y ) .

f o r any f c Cc(G )

f ( y ) d r ( y ) = I f(Y) e-C(Y) d v - l ( Y ) , so t h a t D = ~V e x i s t s and is equal to e -c (v a.e. ). dv -1 (ii) dv -I dv

:>

( i ) The same computation shows t h a t ,

i f u is q u a s i - i n v a r i a n t w i t h

- e -c, then @[mt(f) ~ ~

= @[g ~ ~ t + i ( f ) ]

f o r any f , g c Cc(G,o ).

Second, we consider the case when ~ is i n f i n i t e .

The = - ~IS c o n d i t i o n asserts

t h a t f o r any f c Cc(G,~),

-i@(f~

~

5(f))

Z O.

A f t e r a computation, t h i s becomes Ilfl2c

dv - I > O, where v

Hence, ~ s a t i s f i e s

= I ~Udu(u).

the ~ - KMS c o n d i t i o n i f f

c is non-negative on the support of v - I ,

which is the inverse image under d of the support of ~. c o n d i t i o n f o r u, namely supp ~ c M i n ( c )

But t h i s is j u s t the ~ - KNS

= {u ~ GO : CiGu ~ 0}.

F i n a l l y suppose t h a t G is p r i n c i p a l ,

~ finite

ding to the p o s i t i v e type measure ~, s a t i s f i e s

and t h a t the weight ~, correspon-

the (m,B) KMS c o n d i t i o n . Then f o r any

f , g ~ Cc(G,~ ) and any t ~ ~ , we have

~[~t(f). Using a l e f t

g]

:

~[g . ~t+i(f~.

approximate i d e n t i t y f o r Cc(G,~ ) endowed w i t h the i n d u c t i v e l i m i t

topology,

one gets #(hg) = ~(gh) f o r g ~ Cc(G,{ ) and h ~ Cc(GO). We want to show t h a t the support of u is contained in GO. Suppose t h a t g E Cc(G ) and supp g n GO = @. Since G is p r i n c i p a l , supp g may be covered by open sets U such t h a t d(U) n r(U) = @ . Using a par-

115

n

tition

of the u n i t y , we may w r i t e g = # gi with d(supp g i )

n r(suppgi) = 0. I f we

choose h e Cc(GO) which takes the value 1 on d(supp g i ) and 0 on r(supp g i ) , we have #(gi ) = #(hgi) = #(gi h) = O, hence #(g) = O. Q.E.D. 5.5. Remarks : a.

Since i t

is important to determine a l l KMS weights of a group of automorphisms,

we give the f o l l o w i n g complement f o r ~ = ~. Let # be a weight corresponding to a p o s i t i v e type measure u on G. Then one can show t h a t # s a t i s f i e s the (m,#) KNS c o n d i t i o n only i f suppv c c-1(0) n d-1(Min(c~.In p a r t i c u l a r , one element u,then c - I ( 0 ) n d - 1 ( M i n ( c ) ) i s also reduced to c-l(o)

i n v a r i a n t (1.3.16 ( i v ) ) .

i f Min(c) is reduced to {u} because Min(c) is

Thus there is only one KMS weight at ~, namely, the

p o i n t mass at u. b.

Given an (~,#) KMS weight #, i t

is natural to look at the GNS r e p r e s e n t a t i o n L

i t generates. I t is the r e p r e s e n t a t i o n induced by ~ in the sense of 2.7. I t acts on L2(G,v -1) by l e f t

c o n v o l u t i o n . Let [ be the von Neumann algebra i t generates. There

e x i s t s a unique normal s e m i - f i n i t e weight ~ on £ which extends # in the sense t h a t ~ ( f ) = ~oL(f) f o r f ~ Cc(G,~), and there is a unique automorphism group ~ which extends ~, ato

L ( f ) = L o ~ t ( f ) f o r f ~ Cc(G,~ )-

Let H be the operator of m u l t i p l i c a t i o n

by c on L2(G,v-1). Then ~ is given by

~t(A) = e itH A e- i t H The operator H is i n t e r p r e t e d as the energy operator in t h i s r e p r e s e n t a t i o n . Let us consider the case ~ f i n i t e .

We f i r s t

assume

~ = i . The r e p r e s e n t a t i o n

L is in standard form. I t is the r e g u l a r r e p r e s e n t a t i o n on ~ and appears as the l e f t r e p r e s e n t a t i o n of the g e n e r a l i z e d H i l b e r t algebra introduced in 1.10. In p a r t i c u l a r is the modular group of the f a i t h f u l

normal s e m i - f i n i t e weight ~. The r e l a t i o n between

the modular operator A, which is given by m u l t i p l i c a t i o n

by the R-N d e r i v a t i v e D,

and the energy operator H is A = e-H. In the case when # is a r b i t r a r y but f i n i t e we replace c by ~c and o b t a i n the r e l a t i o n A = e-#H between A and H. In the case B = ~, the r e p r e s e n t a t i o n L is no longer in standard form. The

,

116

~-KMS c o n d i t i o n is p r e c i s e l y the requirement H > O. One says t h a t a vector ~ s L 2 ( G , v -1) has zero energy i f ~(x) = 0 f o r v ground weight ( [ 6 5 ] , d e f i n i t i o n

-1

a.e. x such t h a t c(x) = 0 and t h a t ¢ is a physical

5.2) i f the space of vectors of zero energy is one-

dimensional. A necessary c o n d i t i o n is t h a t u is a p o i n t mass. In f a c t , at u s B i n ( c ) defines a physical ground weight i f f

[u]

the p o i n t mass

n Bin(c) = { u } , where [u]

is the o r b i t of u. c.

Suppose t h a t c s B I ( G , ~ ) ,

c(x) = bor(x) - hod(x). Then we know t h a t ~ is inner.

I t is implemented by the group of u n i t a r i e s Ut(u ) = e i t b ( u ) .

I f we t h i n k of b as the

energy f u n c t i o n , the i n t e r p r e t a t i o n of Bin(c) is c l e a r : Bin(c) = (u s GO : the restriction

of b to [u] reaches i t s minimum at u}. In the general case, we w i l l

call

c the energy cocycle of the system. Given a cocycle c s ZI(G,A), we have defined the C*-dynamical system (C*(G,a),A,~). Our l a s t task in t h i s section is to i d e n t i f y C*(G,a) x

the crossed product C*-algebra

A as the C*-algebra of the skew product, t h a t i s , C ( G * ( c ) , a ) .

Let us

r e c a l l some notations and introduce new ones : G is a l o c a l l y compact groupoid w i t h Haar system (k u) ; a i s a continuous 2-cocycle in Z2(G,~) ; A is a l o c a l l y compact abelian group, noted m u l t i p l i c a t i v e l y multiplicatively

; i t s dual group r = A w i l l

be noted

too ; and c is a continuous 1-cocycle in ZI(G,A). The skew product

G(c) is the l o c a l l y compact groupoid obtained by d e f i n i n g on G x A the m u l t i p l i c a t i o n (x,a)(y,ac(x)) will

= ( x y , a ) and the inverse (x,a) -1 = ( x - l , a c ( x ) ) .

be w r i t t e n ( x , y , a ) instead of ( ( x , a ) , ( y , a c ( x ) ) .

system (Xu,a =

ku x 6a). A cocycle on G l i f t s

A composable p a i r

The groupoid G(c) has the Haar

to a cocycle on G(c), f o r example,

we define a ( x , y , a ) = a ( x , y ) . Let (E,F,m) be a B a n a c h , - a l g e b r a dynamical system, t h a t i s , E is a Banach *-algebra,

r a l o c a l l y compact group and ~ a continuous homomorphism of r i n t o Aut(E)

equipped w i t h the topology of pointwise convergence. Recall t h a t LI(F,E) is the space of E-valued f u n c t i o n s on case, ? is a b e l i a n ) . f *g(~) f*(~)

= If(n)

r i n t e g r a b l e w i t h respect to the Haar measure of F ( i n our

I t is made i n t o a Banach*-algebra with the operations

:

~n[g -1~)]dn,

= f(~-l),

and the norm llfIll= ] l l f ( ~ ) I l d ~. A c o v a r i a n t representation of the system on a H i l b e r t

117 space JC consists of a continuous unitary representation V of r on JC and a norm decreasing nondegenerate representation M of E such that V(~) M(e) V(~)

= M[mc(e)].

A covariant representation (V,M) has an integrated form. Namely, L(f) = f M [ f ( ~

V(~)d~

defines a non-degenerate representation of L on JC. Conversely, i f

E has a bounded

approximate i d e n t i t y , any non-degenerate representation of LI(F,E) is an integrated form and the correspondence is b i j e c t i v e . All this is well known and we refer to [20] for further d e t a i l s . I f E is a C * - a l g e b r a , the crossed product C*-algebra E x

F

is the enveloping C* -algebra of LI(F,E). Recall that we defined the norm

II III on Cc(G,~) by

Itflli = max {sup ~ I f t d x u, sup I I f l d Z u ) U

U

It is a *-algebra norm on Cc(G,o). We denote the completion of Cc(G,~) in the norm II III by LI(G,o). One annoying problem with this Banach * - a l g e b r a is the existence of a bounded approximate i d e n t i t y . I t can be established without d i f f i c u l t y

in the r -

discrete case (take a bounded approximate i d e n t i t y for C*(GO)) and when G is a transformation group (take the pointwise product hie i , where e i is the c h a r a c t e r i s t i c function of a symmetric neighborhood of the i d e n t i t y of the group, normalized for the l e f t Haar measure, and e i a bounded approximate i d e n t i t y for C*(GO)), but I don't know i f i t always exists in the general case. Note that, as a Banach space, LI(G(c),~) is Co(A,LI(G,o)), the space of Ll(G,o)-valued continuous functions on A which vanish at i n f i n i t y . 5.6. Lemma :

Let E be a separable Banach space, F a l o c a l l y compact abelian group

and ~ a continuous homomorphism of r into the group of isometries of E equipped with the topology of pointwise convergence. Then

f

~ ~(a) = fF

~-iEf(~)]

(~,a) de

defines a norm-decreasing l i n e a r map with dense range from LI(F,E) into CO(F,E). Proof :

Clearly, f(a) is well defined and llf(a)II

~ NfII. By the Lebesgue dominated

convergence, f is a continuous function from F to E. I f f is a decomposable element n

of L I ( r , E ) , that i s , an element of the form f(~) =

f i ( ~ ) e i , where f i c L1(r) and

ei ~ E for i = 1, . . . . n, then f vanishes at i n f i n i t y .

Since decomposable elements

are dense in L I ( r , E ) , the map sends LI(F,E) into CO(F,E). We want to show that i t has

118

dense range. Note that the map 6, defined by 6(f)(~) of L I ( r , E ) . If(C)

= ~ -l[f(~)]

Therefore, i t suffices to consider the map f

(~,a)d~. Since the Fourier transform from L l ( r )

is an isometry

÷ f, where f(a) =

into CO(r ) has dense range, every

decomposable element of CO(~,E ) lies in the range closure of the map f

÷ f.

Since

decomposable elements are dense in CO(~,E), the range closure is Co(r,E ). O.E.D. 5.7. Theorem :

Let G, o, A, c and m be as above. Assume that LI(G,a) has a bounded

approximate unit. Then the crossed-product C*-algebra to the C*-algebra Proof :

C*(G(c),a)

C*(G,o) x

A is isomorphic

of the skewproduct.

Since the automorphism

~ of Cc(G,a ), given by ~ ( f ) ( x )

= (~,c(x))f(x),

preserves the [[ III norm, i t extends to an automorphism of LI(G,a). The continuity of ~ : r ÷ A u t ( L l ( G , ~ ) )

is established as in 5 . 1 . ( i i i ) .

to Co(A,LI(G,a)) = LI(G(c),~) defined by f

By 5.6, the map from LI(A,LI(G,a))

÷ f(a) = f r ~ - l [ f ( ~ ) ] ( ~ ' a ) d ~

decreasing and has dense range. It is a straightforward is a * - a l g e b r a f(x,a)

=

is norm-

computation to check that i t

homomorphism. Let us j u s t write down the relevant formulas ~f(x,~)(~,ac(x))

d~

for f,g ~ Cc(G x A ) c L I ( A , L I ( G , ~ ) ) , f * g (x,~) = I f f(y,n) g(y-lx,n-Z~)

(n,c(y-Zx))a(Y,y-lx)d~r(X)(y)dn

f * (x,~) = ~(x - I , ~-1) ~-(x,x-Z) (~,c(x)) and for f,g E CO(A,Cc(G))CLI(G(c),a), f . g (x,a) = ] f ( y , a ) g ( y - l x , a c ( y ) )

a(y,y-lx)

d~r(X)(y)

f * (x,a) = ~ ( x - l , a c ( x ) ) T ( x , x - 1 ) . Composing with the homomorphism of LI(G(c),~) decreasing)*-algebra

into C* (G(c),~), we obtain a (norm-

homomorphism ~ from LI(A,LI(G,a))

dense range. I f L is a (non-degenerate)

= ll~M[f(~)Iv(~)d~ll ~llf(~) 11d~ = Nfll I

is a non-

There exists a covariant representation

is the integrated form. By d e f i n i t i o n

C (G,a), M decreases i t s C*-norm and we obtain the estimate IILo~(f)ll

which has

representation of C* (G(c),a), Lo~

degenerate representation of LI(A,LI(G,~)). (V,M) of (A, LI(G,a), of which Lo~

into C * ( G ( c ) , a )

of

119 where ]] I{I is the norm of LI(A,C* (G,a)). Therefore, LoT extends to a representation of LI(A,C*(G,o)) and ipso facto to a representation of i t s enveloping C*-algebra C*(G,o)×m A. We have I]Lo~(f) II # [IfI[, where [I IIis the norm of C*(G,o) x A. We conclude that ll~(f) II ~ C*(G(c),o).

Ilfll and that

~ extends to a *-homomorphism from C * ( G , o ) x A to

I t is onto, because i t s range is dense and closed. Let us show that i t

is one-to-one, or, equivalently, isometric. Let L be the representation of C*(G,o) x A induced by the representation M of C* (G,~). We w i l l assume that M is the integrated form (cf. theorem 1.20) of the {-representation (~,T~,M) of G. H = r(~) (the space of square integrable sections o f ~ )

Let

be i t s representation space.

By d e f i n i t i o n , L acts on L2(A;H) by k(f)~(y) = f r M [ ~ y - l ( f ( ~ ) ) ]

~(~-Iy)d~

where f ~ LI(A;C*(G,~)) and @cL2(A;H). Let us consider the following o-representation (~xx~JC,[) of G(c) : k is the Haar measure of A (we have observed in 3.8 that M×kis q u a s i - i n v a r i a n t ) , Jgu, a =~(Ju ; and [(x,a) is given by [ ( x , a ) = M(x). Its [(f)@(u,a) = f f ( x , a )

: JC(d(x),ac(x))

÷JC(r(x),a)

integrated form acts on F(;}C) by

[(x,a)@(d(x),ac(x)) D-l/2(x)dxU(x),

for f ~ Cc(G(c),o ), @~F(JC), where D is the modular function of ~. We may i d e n t i f y F(JC) with L2(A,H) in an obvious fashion, where H = F(~), and we may define the Fourier transform ~-from L2(A,H) to L2(A;H) by ~-@(a) =f@(y)(y,a) dy. Of course, ~ - i s an isometry. I t is then a straightforward compu~tion to check that ~ok(f) = [ ( f )

o~-

for any f ~ Cc(A x G), where f = ~ ( f ) . The relevant formulas are L(f)~(u,y) = f f ( y , c ( x ) )

f(x,~) M(x)~(d(x),~-Iy) D-1/2(x) d~U(x)d~,

[(f)@(u,a) = f f ( x , a )

M(x)@(d(x),ac(x)) D-1/2 (x) d~U(x) ,

f(x,a)

=ff(x,~)

(~,ac(x)) dg, and

~-@(u,a)

:f@(u,y)

(~,a) dy .

This shows that IIL(f)H ~

II~(f)II for every f ~ LI(A,C*(G,o)) and every induced

representation L. Since A is abelian, the reduced norm on LI(A,C* (G,~)) coincides with the C*-norm ([70], proposition 2.2). Hence llfll ~

II~(f)]I.

Q.E.D.

120 5.8. C o r o l l a r y : a

Let G be an r - d i s c r e t e amenable p r i n c i p a l groupoid w i t h Haar system,

E Z2(G,T), A a l o c a l l y compact abelian group and c E ZI(G,A). Then the asymptotic

range R (c) of c coincides w i t h the Connes spectrum

r(m) of the corresponding

automorphism group m on C*(G,~). Proof :

We i d e n t i f y the crossed product C * - a l g e b r a C * ( G , o )

×

A and C* (G(c),~) .

The canonical action of A on the skew product G(c), s ( a ) ( x , b ) = ( x , a b ) , defines an action on A on C * ( G ( c ) , ~ ) ,

Ba(f)(x,b)

= f(x,a-lb).

Thus C*(G(c),~),A,~)

is nothing

but the dual system of (G (G,~),A,m). The Connes spectrum r(m) can be characterized as ( [ 6 0 ] , 8.11.8) r(m) = {a c A : J n Ba(J ) # {O}for C

(G(c),o)}.

every non-zero ideal J of

Using the correspondence 4.6 between ideals of C*(G(c),~) and i n v a r i a n t

open subsets o f the u n i t space o f G(c), the amenability of G(c) and 1.4.10, one gets the conclusion.

O.E.D.

5.9. Remark : We have r e s t r i c t e d our a t t e n t i o n to automorphism groups of C*(G,~) which stem from a cocycle c c ZI(G,A). Another kind of automorphism group which leaves C * ( G O) i n v a r i a n t is given by a continuous action of a group A by automorphisms of G leaving the Haar system i n v a r i a n t and a s i m i l a r study can be done.

CHAPTER I I I

SOME EXAMPLES

We shall give here two kinds of examples of r - d i s c r e t e groupoids with Haar system. Our f i r s t

v example results from the observation by S t r a t i l ~ and Voiculescu ([69], ch. I ,

§ I , page 3) that approximately f i n i t e - d i m e n s i o n a l C * - a l g e b r a s ( f o r short AFC*-algebras) could be diagonalized. This fact had already been used in a p a r t i c u l a r case be o

Garding and Wightman in [34] to construct i n f i n i t e l y

many non-equivalent i r r e d u c i b l e

representations of the anticommutation r e l a t i o n s . In the terminology of 2.4.13, t h i s can be rephrased by saying that AF C~-algebras have

Cartan subalgebras. Thus, an

AF C * - a l g e b r a is the C * - a l g e b r a of an r - d i s c r e t e p r i n c i p a l groupoid. The groupoids which arise in that fashion (we call them AF) are studied in the f i r s t

section. They

have also been considered, in a form where the emphasis was on the ample group rather than on the groupoid, by Krieger in [52]. Our second example is given by the C * - a l g e bras generated by isometries introduced and studied by Cuntz in [15]. We show that these C*-algebras may be w r i t t e n as groupoid C * - a l g e b r a s . The corresponding groupoids, which are described in the second section and which we c a l l On , are not p r i n c i p a l . In both cases, the description of the C * - a l g e b r a in terms of a groupoid is used to discuss the existence of KMS-states with respect to some automorphism groups.

1.

Approximately F i n i t e Groupoids.

The simplest examples of r - d i s c r e t e principal groupoids are, on one hand, the l o c a l l y compact spaces (corresponding to the equivalence r e l a t i o n u ~ v i f f

u = v)

122 and, on the o t h e r , the t r a n s i t i v e

p r i n c i p a l groupoids on a set o f n elements, where

n = 1,2 . . . . ~ (corresponding to the equivalence r e l a t i o n u ~ v f o r every u and v) w i t h the d i s c r e t e topology. By means o f elementary o p e r a t i o n s , we may combine them to o b t a i n o t h e r examples. The product o f two groupoids is defined in the obvious fashion. I f the groupoids are t o p o l o g i c a l , then the product is given the product topology and i f each o f the groupoids is endowed with a Haar system, the product is given the product Haar system ; u. (Ul,U2) uI u2 e x p l i c i t l y i f {~i I } is a Haar system f o r Gi, i = 1,2, then {~ = ~1 x ~2 } is a Haar system f o r G1 x G2" Another o p e r a t i o n makes sense in the category o f groupoids ; t h i s is the d i s j o i n t union. Let Gi be a groupoid, with i = 1,2 ; then define G = G1 • G2 as the settheoretical disjoint

union o f G1 and G2 with the groupoid s t r u c t u r e given by the rules

"x and y are composable in G i f f

they belong to the same Gi and are composable in Gi

and t h e i r product in G is equal to t h e i r product in Gi" and " i f x belongs to Gi ,

its

inverse in G is equal to i t s inverse in Gi". I f the groupoids Gi are t o p o l o g i c a l , u. then t h e i r d i s j o i n t union is given the d i s j o i n t union topology and i f {~i I } is a Haar system f o r Gi, i = 1,2, then {~u}, where ~u = I~

i f u ~ Gi O, is a Haar system

f o r G. One can d e f i n e in a s i m i l a r fashion the d i s j o i n t

union o f a sequence o f grou-

poids. A l a s t o p e r a t i o n which we need here is the i n d u c t i v e l i m i t . restricted definition,

sufficient

We give here a

f o r our purposes. Suppose t h a t the groupoid G is

the union o f an increasing sequence o f subgroupoids Gn, which a l l have the same u n i t space as G ; then we say t h a t G is the i n d u c t i v e l i m i t o f the sequence (Gn). I f G is t o p o l o g i c a l , we r e q u i r e t h a t Gn be an open subgroupoid o f G. I f {~u} is a Haar system f o r G, we consider the Haar system restriction

of

u

to r n l ( u ) .

{~}

on Gn such t h a t

~n u

is

the

Conversely, suppose t h a t the Gn s are t o p o l o g i c a l

groupoids such t h a t Gn is open in Gn+1 and i t s topology is the topology induced from Gn+I . Then, the i n d u c t i v e l i m i t topology, where a set V is open i f f in Gn f o r every n

makes G i n t o a t o p o l o g i c a l groupoid

then so is G. F i n a l l y ,

u i f each Gn has a Haar system {~n }

compatible, in the sense t h a t

~n u is the r e s t r i c t i o n

of

V n Gn is open

I f the ~n s are l o c a l l y compact, and i f these measures are

u ~n+l

to r~

l(u )

then there

123

e x i s t s a unique Haar system { u} such t h a t ~u is the r e s t r i c t i o n n

o f ~u to r n l ( u )

Let us note t h a t these operations preserve a m e n a b i l i t y ( d e f i n i t i o n

2.3.6).

Let

us show, f o r example, t h a t the i n d u c t i v e l i m i t G of a sequence (Gn) of amenable groupoids is amenable. Let K be a compact subset o f G and E a p o s i t i v e number. Since I

the Gn s are open, K is contained in some Gn. Since Gn is amenable, there e x i s t s f ~ Cc(Gn) such t h a t

If*

*

f ( x ) - 11 ~ ~ f o r x E K (and j I f ( x ) I

2u dAn bounded by 2).

Then f E Cc(g ) and s a t i s f i e s the same c o n d i t i o n in G. 1.1. D e f i n i t i o n :

Let G be an r - d i s c r e t e groupoid. We say t h a t G is an elementary

groupoid o f type n (n = 1,2 . . . . . ~) i f

it

is isomorphic to the product o f a second

countable l o c a l l y compact space and o f a t r a n s i t i v e p r i n c i p a l groupoid on a set of n elements. We say t h a t G is an elementary groupoid i f

it

is the d i s j o i n t

union o f a sequen-

ce o f elementary groupoids o f Gi o f type n i . We say t h a t G is an approximately elementary (AE! groupoid i f

it

is the i n d u c t i v e

l i m i t of a sequence of elementary groupoids. We say t h a t G is an approximately f i n i t e elementary and i t s u n i t space is t o t a l l y

(AF) groupoid i f

it

is approximately

disconnected.

1.2. Remarks : A l l these groupoids are p r i n c i p a l and amenable since these p r o p e r t i e s are preserved under product, d i s j o i n t

union and i n d u c t i v e l i m i t .

They have the coun-

t i n g measures as Haar system. The o r b i t s o f an elementary groupoid of type n have the same c a r d i n a l i t y n. However there e x i s t r - d i s c r e t e p r i n c i p a l groupoids, a l l o r b i t s o f which have the same c a r d i n a l i t y n, which are not elementary o f type n. An example is given by the equivalence r e l a t i o n on the c i r c l e which i d e n t i f i e s

two points l y i n g on the same diameter.

The u n i t space of t h i s groupoid is connected, w h i l e the u n i t space of an elementary groupoid o f type 2 has a t l e a s t two components. The terminology o f elementary groupoid does not agree with the d e f i n i t i o n (4.1.1) in [ 1 ~ )

o f an elementary C * - a l g e b r a . Only t r a n s i t i v e

give elementary C* -algebras.

p r i n c i p a l groupoids

124 1.3. P r o p o s i t i o n : (i)

Let G be an elementary g r o u p o i d . Then, f o r every G-module bundle A (not

n e c e s s a r i l y a b e l i a n ) , every cocycle c c ZI(G,A) i s i n n e r , ( t h a t i s , (ii)

i s a coboundary).

Let G be an a p p r o x i m a t e l y elementary g r o u p o i d . Then, f o r every G-module

bundle A (not n e c e s s a r i l y a b e l i a n ) , every cocycle c E ZI(G,A) i s a p p r o x i m a t e l y i n n e r in the sense t h a t i t

can be approximated by coboundaries u n i f o r m l y on the compact

subsets o f G. (iii)

Let G be an a p p r o x i m a t e l y elementary g r o u p o i d . Then, f o r every a b e l i a n G-

module bundle A and every n ~ 2, Hn(G,A) = 0. Proof : (i) locally

We w i l l

show t h a t an elementary groupoid i s ( c o n t i n u o u s l y ) s i m i l a r to a

compact space. Since a l o c a l l y

homology, t h i s w i l l

compact space (as a groupoid) has t r i v i a l

prove the a s s e r t i o n .

I t s u f f i c e s to c o n s i d e r the case o f an

elementary groupoid o f type n, o f the form G = X x I , where X is a l o c a l l y n space and I n

the t r a n s i t i v e

co-

groupoid on {1 . . . . . n}. Then, a s i m i l a r i t y

compact

between

G and X is given by : X × I n ÷ X and (x,(i,j))

~ : X ÷ X x In

~ x

x ~ (x,(1,1))

because #o~ = id X and ~ o ¢ ( x , ( i , j ) ) X x {1 . . . . . n}

÷ X x I

(x,i)

~ (x,(i,i)),

(ii)

= e(x,i)idG(X,(i,j))e(x,j)

e is the map

n

Let G be the i n d u c t i v e l i m i t

and l e t c c Z I ( G , A ) . By ( i ) ,

-1 where

o f a sequence o f elementary groupoids Gn

the r e s t r i c t i o n

ClG n

o f c to Gn i s a coboundary on Gn,

hence may be extended to a coboundary c n on G. Since every compact subset o f G i s contained i n some Gn, (Cn) converges to c u n i f o r m l y on the compact subsets o f G. (iii)

Write G as i n c r e a s i n g union o f a sequence o f elementary groupoids Gi ,

Let ~ e Zn(G,A), w i t h n ~ 2. I t s r e s t r i c t i o n However Zn(Gi,A) =

to Gi ,

(0) f o r n ~ 2, since Zm(Gi,A) =

~i'

belongs to Zn(Gi,A).

Bm(Gi,A) f o r m > 1. Thus o = O. Q.E.D.

125

An e s s e n t i a l f e a t u r e o f an a p p r o x i m a t e l y elementary groupoid G i s t h a t i t (c,~) KMS measures f o r every c c ZI(G,~) and every

has

B E [ - ~ , + ~ ] , provided t h a t i t s

u n i t space i s compact. 1.4. Lemma :

Let G be a l o c a l l y compact groupoid w i t h Haar system and l e t c be a

coboundary in BI(G,J~). (i)

I f GO i s compact, then (c,~) KMS p r o b a b i l i t y

(ii)

measures e x i s t .

I f t h e r e i s a ( c , ~ ) KMS measure f o r some ~ c ~ ,

then t h e r e are ( c , B ' )

KMS

measures f o r every 8' E ~ . (iii) B e~,

I f GO is compact and i f then t h e r e are ( c , B ' )

Proof :

t h e r e is a (c,B) KMS p r o b a b i l i t y

KMS p r o b a b i l i t y measures f o r every

measure f o r some 8' E [ - ~ , + ~ ] .

Let us w r i t e c ( x ) = h o r ( x ) - hod(x) where h i s a continuous f u n c t i o n on GO.

(i)

The set Minh o f the p o i n t s o f GO where h reaches i t s minimum is non-empty

and contained in Minc. The point-mass a t such a p o i n t i s a ( c , = ) KMS p r o b a b i l i t y measure. (ii)

I f p i s a (c,B) KMS measure, then f o r every 8' e ~ , the measure ~' given by

d~'(u) : exp[-(~'-B) is a ( c , # ' )

h(u)]

d~(u)

KMS measure. For, i f v' = I ~ U d u ' ( u ) :

d r ' -1 d~'

d~l (x) exp [ ( ~ - ~ ' ) h o d ( x ) ] d~

(x) = e x p [ - ( B ' - ~ ) h o r ( x ) ] = exp [ - B c ( x ) ] .

(iii)

I f u, as above, i s f i n i t e

and i f GO i s compact, u' i s a l s o f i n i t e . Q.E.D.

1.5. P r o p o s i t i o n :

Let G be an a p p r o x i m a t e l y elementary groupoid w i t h compact u n i t

space. Then i t admits (c,B) KMS p r o b a b i l i t y measures f o r every c E ZI(G,~) and every B E

[-~,+~].

Proof : Since elementary groupoi~s w i t h compact u n i t space have f i n i t e measures, they have (c,B) KMS p r o b a b i l i t y E~

measures f o r every c and every 8. Fix

andc c Z I ( G , ~ ) . Write G as the i n d u c t i v e l i m i t

groupoids and l e t c

n

be the r e s t r i c t i o n

invariant

o f a sequence (Gn) o f elementary

o f c to G . For each n, t h e r e e x i s t s a n

126

p r o b a b i l i t y measure ~n whose modular f u n c t i o n with respect to Gn is be a l i m i t

e

-6c n

Let

p o i n t o f the (Un)'S f o r the weak , - t o p o l o g y o f the dual o f the space

of continuous functions on GO. I f Un ÷ u' then Vn ÷ ~

and ~n-1 ÷ v - I f o r the weak

* - t o p o l o g y o f the dual o f Cc(G). Therefore, f o r every f E Cc(G), Jfdv - I = l i m

Jfd~nl=

lira

J fe6Cnd~n = J fe~Cd~.

This shows t h a t the modular f u n c t i o n o f ~ e x i s t s and is e -~c. The statement about i n f i n i t e

6 r e s u l t s from 1.3.17. Q.E.D.

1.6. Example : The I s i n g model. The points o f Z = Z v are the sites o f a crystal where v is an i n t e g e r . Each s i t e has a spin up ( - I ) the l a t t i c e

lattice

or down ( - i ) .

is given by a f u n c t i o n u o f Z i n t o { - 1 , + I } .

{ - 1 , + I } Z is given the product topology ; i t w i l l Two c o n f i g u r a t i o n s are e q u i v a l e n t i f f

of dimension v, A configuration of

The space o f c o n f i g u r a t i o n

be the u n i t space GO o f the groupoid.

they d i f f e r

at most f i n i t e l y

many s i t e s .

The corresponding p r i n c i p a l groupoid is noted G. We choose an increasing sequence (Zn) o f f i n i t e

subsets o f the l a t t i c e

such t h a t Z = u Zn and d e f i n e the subgroupoid

Gn by the equivalence r e l a t i o n : "two c o n f i g u r a t i o n s are e q u i v a l e n t i f they agree outside Zn . Then Gn is an elementary groupoid of the form { - I , + I } Z\Zn × l[Zn] and G =uGn. We give to G the i n d u c t i v e l i m i t topology. Thus G is an AF groupoid. The dynamics o f the system are described by the f o l l o w i n g energy cocycle c ~ Z I ( G , R ) given by the expression c(u,v) = .~.

J(i,j)

{(i - uiuj)

- (1 - v i v j ) } ,

l~J

where J depends on the nature o f the i n t e r a c t i o n . The sum is in f a c t f i n i t e there are f i n i t e l y

since

many non zero terms.

From 1.5, the system has I~MS states f o r every

~. The ground states are the

measures which l i v e on {u ~ Go : uiu j = i whenever J ( i , j )

# 0}. In p a r t i c u l a r ,

the

c o n f i g u r a t i o n s (u i = +i f o r every i ) and (u i = -1 f o r every i ) are physical ground states. Some r e s u l t s , depending on ~ and on J, are known above the existence o f d i s t i n c t KMS states at a given B.The parameter B is i n t e r p r e t e d as the inverse temperature

127 and ~4S states are e q u i l i b r i u m states. Coexistence of d i s t i n c t KMS states means the existence of several "phases". I f the l a t t i c e were f i n i t e ,

G would be f i n i t e ,

c inner

and there would be one and only one KMS state f o r every 6. The interested reader w i l l f i n d a review of these r e s u l t s as well as a bibliography in the A.M.S. a r t i c l e by J. Fr~hlich [33]. We turn now to the properties of the skew-product G(c) where G is approximately elementary (or f i n i t e ) . 1.7. Proposition :

Let G be a l o c a l l y compact groupoid, A a l o c a l l y compact group

and c a cocycle in ZI(G,A). (i)

I f G is approximately elementary, then the skew product G(c) of G by c is

approximately elementary. (ii)

I f G is approximately f i n i t e and A is t o t a l l y disconnected, then G(c) is

approximately f i n i t e . Proof : (i)

I f c is a coboundary, c(x) = bor(x) (bod(x)) - I

Then, the map from G x A

to G(c) sending (x,a) to (x, a(bor(x)) -1) is an isomorphism of groupoids, when G × A is given the product structure and where A is viewed as a l o c a l l y compact space. Therefore, i f G is elementary, G(c) is also elementary for every c c ZI(G,A). Suppose now that G = uGn with Gn elementary. Let c ~ Z1 (G,A) and l e t c n be i t s r e s t r i c t i o n to Gn. Then G(c) = UGn(Cn) and Gn(Cn) is elementary. Thus, by d e f i n i t i o n , G(c) is approximately elementary. (ii)

From the f i r s t

part, we know that G(c) is approximately elementary. Moreover

i t s u n i t space GO × A is t o t a l l y disconnected. Hence i t is approximately f i n i t e . Q.E.D. Remark : in [ 4

This l a s t proposition gives a p a r t i a l answer to a question B r a t t e l i asks

(problem 2, page 35). I f (~t,G,~) is a C* -dynamical system with.,{ AF and G

compact, is the crossed product a l g e b r a ~ x G necessarily AF ? This is so i f G is abelian and the action is given by a cocycle as in 2.5.1.

128 The crossed-products of UHF algebras by product-type actions studied by B r a t t e l i in [9] are a p t l y described in terms o f groupoids. Let (Xi) be a sequence o f f i n i t e d i s c r e t e spaces and l e t

X =~X i be t h e i r product, with the product t o p o l o g y . The

equivalence r e l a t i o n ~ on X, where u ~ v i f f

ui = v i for a l l

but a f i n i t e

number of

i n d i c e s , defines a p r i n c i p a l groupoid G. I f the sequence is indexed by N we may def i n e the groupoid Gi = { ( u , v )

~ G : uj = vj f o r j ~ i } , w h i c h

is elementary. As in

example 1.6, G =UG i is made i n t o a t o p o l o g i c a l groupoid which is AF. Since every p o i n t o f GO = X has a dense o r b i t , to such a groupoid G w i l l

G is minimal.A t o p o l o g i c a l groupoid isomorphic

be c a l l e d a Glimm groupoid, because, as we shall see, i t s

C ~ - a l g e b r a is a UH~or Glimm, algebra. Let A be an a b e l i a n l o c a l l y compact group. A cocycle c ~ ZI(G,A) w i l l of product type i f

be said

i e is o f the form

c(u,v) = ~ ci(ui,vi)

where c i c ZI(Gi,A)

where Gi is the t r a n s i t i v e groupoid on the set Xi . We may w r i t e ci(ui,vi)

= bi(ui)

- bi(vi)

with bi f u n c t i o n from Xi i n t o A. We l e t Ci = c i ( G i ) = Bi - Bi where Bi = b i ( X i ) . may assume t h a t 0 ~ Bi .

We

Let us note t h a t , by the d e f i n i t i o n o f the topology o f G as

i n d u c t i v e l i m i t topology, a cocycle of product type is continuous. 1.8.

Proposition :

Let G be a Glimm groupoid, A an a b e l i a n l o c a l l y compact group

and l e t c be a cocycle in ZI(G,A) of product type as above. ( i ) The asymptotic range o f c is R (c) = j ~ (ii)

I t s T-set is T(c) ={~ ~ A : V ~ > O,

J :

(i!j_ Ci)" I~( i>_j

(iii)

The cocycle c is a coboundary i f f

Bi ) - I f

<_ e}.

f o r every neighborhood V of 0 in A, there

e x i s t s j such t h a t

~ Bi is contained in V. i~j ( i v ) The asymptotic range o f c at u is R~(c) =

f~ join

(i!j_ Bi - bi)

where b i = b i ( u i ) . Proof :

The assertions ( i ) ,

assertion (iii)

(ii),

and ( i v ) r e s u l t from the d e f i n i t i o n

1 . 4 . 3 . The

r e s u l t s from p r o p o s i t i o n 1.4.8. We have to check t h a t the hypotheses

o f t h i s p r o p o s i t i o n are s a t i s f i e d . The u n i t space o f G is compact and G admits a cover o f continuous G-sets, namely, the sets

129

• "" s= {~a,ui;,~o~al,u1~j'" "' '"

~ G : a ~

i-ll_T X ~ , u i 1 J

~

l-[

j~i

X~

}

J

where i is an i n t e g e r and { a b i j e c t i o n of ~c~z X< onto i t s e l f . J 1 Q.E.D. Bratteli

points out in [ 9 ] t h a t the s i m p l i c i t y of the crossed-product algebras

he considers depends h e a v i l y on the s t r u c t u r e of the group A. This is summarized in the f o l l o w i n g p r o p o s i t i o n . 1.9. Proposition :

Let G be a Glimm groupoid, A an abelian l o c a l l y compact group and

l e t c be a cocycle in ZI(G,A) of product type. (i)

I f A is compact and R (c) = A, then G(c) is minimal

(ii) Proof :

I f A can be ordered, then G(c) is not minimal. The assertion ( i ) r e s u l t s d i r e c t l y

from 1.4.16 ( i i ) .

To prove the second a s s e r t i o n , we use the n o t a t i o n given above. We may choose b i so t h a t Bi = b i ( X i ) is contained in the p o s i t i v e cone P of A and b i is non-decreasing when Xi = { 0 , I . . . . . n i }

has i t s usual order. Let 0 and 1 denote r e s p e c t i v e l y

the sequences 0 = (0,0 . . . . . ) and 1 = ( n l , n 2 . . . . ) in X =4"-FXi . Then the asymptotic range at 0 of c

'

R~(c) =

~ J~

(

Z B i ) , is contained in the p o s i t i v e cone of A while the i~j

asymptotic range at i of c, Rl(c) = .N ( ~ Bi) is contained in the negative cone JqN i~j of A, -P. By 1.4.14 ( i ) , f o r every a e A, the points (O,a) and (1,a) do not have a dense o r b i t .

Therefore, G(c) is not minimal.

Q.E.D. 1.10. Example : Let us f i r s t relations).

The gauge automorphism group of the CAR algebra. define the CAR groupoid (CAR stands f o r canonical anticommutation

I t is a Glimm groupoid isomorphic to the groupoid o f the Ising model.

The p o s i t i o n s of a system of fermions are labeled by a countable set of i n d i c e s , say N- The u n i t space of the groupoid is X = i~T~Xi WnereXi={O,1}.A c o n f i g u r a t i o n u=(u i ) in X t e l l s

i f there is a fermion at the place i.As before, G is the p r i n c i p a l groupoid

given by the equivalence r e l a t i o n ~, where two c o n f i g u r a t i o n s are e q u i v a l e n t i f f they d i f f e r

at at most a f i n i t e

number of places. We shall see t h a t i t s C * - a l g e b r a

130 is the C * - a l g e b r a of the canonical anticommutation r e l a t i o n s (see [8] or [29] page 269). The gauge automorphism group is defined by the product cocycle c c ZI(G,Z), called the "number" cocycle, given by c(u,v) =

~ ui - v i .

The number cocycle counts the number o f p a r t i c l e s by which the configurations u and v d i f f e r . With above notations, Bi = {0,1} and R (c) = Z . Let us define next the GICAR groupoid (GI stands f o r gauge i n v a r i a n t ) . I t is the subgroupoid c - l ( o ) .

In other words, i t corresponds to the equivalence r e l a t i o n m,

where two configurations u and v are equivalent i f f

they d i f f e r at at most a f i n i t e

number of places and have the same number of p a r t i c l e s (in the sense that c(u,v) = 0). Its C * - a l g e b r a is the subalgebra o f fixed points of the gauge automorphism group ; i t is c a l l e d the GICAR algebra. I t results from 1.4.17 that the GICAR groupoid is i r r e d u c i b l e . More information about i t w i l l be given a f t e r we introduce the dimension group of an AF-groupoid. F i n a l l y , l e t us consider the skew-product groupoid G(c). By 1.7 i t is an AF-groupoid. I t is i r r e d u c i b l e (by 1.4.13) but not minimal (by 1.9). The remainder of this section is devoted e x c l u s i v e l y to topological groupoids which admit a base o f open sets consisting of compact open G-sets. A f t e r a few d e f i n i t i o n s , we shall study the example of AF-groupoids. Let G be a topological groupoid which admits a base o f compact open G-sets. Its ample semi-group ~ has been defined (1.2.10) as the inverse semi-group of i t s compact open G-sets. The idempotent elements of ~ are compact open subsets of the u n i t space GO of G. They form a generalized Boolean algebra ~0 (that i s , a Boolean algebra without the assumption that a greatest element e x i s t s ) . We define the f o l l o w i n g equivalence r e l a t i o n on ~ 0 . We shall declare e and f equivalent, and w r i t e e iff

f,

there exists s ~ (~ such that e = r(s) and f = d(s), where r(s) = ss -1 and

d(s) : s - l s . Using terminology common to the theory of von Neumann algebras, one can make the following d e f i n i t i o n .

131 1.11.

Definition

:

Let G be a t o p o l o g i c a l

groupoid which has a base o f compact

open G-sets and l e t ~ be i t s ample semi-group. We say that an idempotent element e of ~ i s

finite

i f f o r any idempotent element f , the r e l a t i o n e ~ f < e implies f = e.

We say t h a t G is of f i n i t e infinite

type i f every idempotent element of ~ is f i n i t e

type otherwise.

We may define on ~0 e

e1

and of

the r e l a t i o n e < f i f f

f . We may also define a p a r t i a l

elements e and f can be added i f f

there e x i s t s e I and f l

such t h a t

a d d i t i o n in ~ 0 , where two idemDotent

they are d i s j o i n t

We denote by D(G) the o f equivalence classes

~0/~

and e + f is the union of e and f . and by D the q u o t i e n t map of 9 0

onto D(G). We provide D(G) w i t h the r e l a t i o n D(e) ~ D(f) i f f a d d i t i o n , where two classes D(e) and D(f) can be added i f f elements e I and f l and then D(e) + D(f) = D(e I + f l ) .

e < f and w i t h a p a r t i a l they contain d i s j o i n t

I f G is of f i n i t e

type, the

r e l a t i o n < is an order r e l a t i o n . 1.12. D e f i n i t i o n

:

Let G be a t o p o l o g i c a l

groupoid which admits a base of compact

open G-sets and l e t ~ be i t s ample semi-group. Assume t h a t G is of f i n i t e i t s dimension range

is the set D(G) =(~0/~

type. Then,

w i t h the order s t r u c t u r e and the p a r t i a l

a d d i t i v e s t r u c t u r e defined as above. I t can be shown ( c f .

[ 2 ~ and [27] in the AF case) t h a t the dimension range D(G)

of G can be embedded in a unique fashion as a generating upward d i r e c t e d h e r e d i t a r y subset of a d i r e c t e d ordered abelian group, c a l l e d the dimension group of G and denoted by Ko(G). The property of being of f i n i t e unions and i n d u c t i v e l i m i t s .

type is preserved under f i n i t e

products, d i s j o i n t

I t can be shown t h a t

Ko(G 1 x G2) = Ko(G1) ® Ko(G2) , w i t h p o s i t i v e cone generated by

K;(G I)

K (G 2) ; n

D(G1 x G2) = {~ mi x n i ; n

n

n

, mi ~ D(G1), n i ~ D(G2), ~ mi ~ D(G1), ~ n i ~ D(G2)}

132

KO( e Gn) = ~Ko(Gn) , where

• Gn is the d i s j o i n t

union of a sequence (Gn) and ~Ko(Gn) is the d i r e c t sum

of the ordered abelian groups Ko(Gn) ; k D( • Gn) = {~ mi : k ~ , m o(ll_~ Gn) = ~

mi c D(Gi)},

mo(Gn) ,

where lim G is the i n d u c t i v e l i m i t the i n d u c t i v e l i m i t

of an increasing sequence (Gn) and l i m Ko(Gn) is

of the ordered abelian groups Ko(Gn) ; and

D(li_~m Gn) = u D(Gn) . An ordered abelian group w i l l if it

be c a l l e d an E l l i o t t

group ( c f .

[25],

[27] and [28]]

is the i n d u c t i v e l i m i t of a sequence of ordered groups, each isomorphic to the

d i r e c t sum of f i n i t e l y an E l l i o t t

many copies of T w i t h

group is preserved under f i n i t e

i t s usual order. The property of being tensor products, countable d i r e c t sums and

countable i n d u c t i v e l i m i t s . For example, the dimension group of a second countable t o t a l l y l o c a l l y compact space X is an E l l i o t t

disconnected

group. Indeed, the dimension range of X is the

(generalized) Boolean algebra of i t s compact open subsets, which may be w r i t t e n as an increasing union of a sequence of f i n i t e

Boolean algebras ~ n and i t s dimension

group is the group of continuous functions with compact support of X i n t o Cc(X,Z ) =

Z

{ f e C c ( X , Z ) : suppf e ~ n }, with i t s usual order. The dimension range

of the t r a n s i t i v e

groupoid on a set of n elements, where n = 1,2 . . . . ,=, is { 0 , 1 , . . . , n }

and i t s dimension group, which is Z ,

is an E l l i o t t

group o f an AF groupoid is an E l l i o t t

group.

group. Therefore, the dimension

The importance of the dimension ranges and of the dimension groups in the study of AF groupoids is given by the f o l l o w i n g p r o p o s i t i o n . The f i r s t e s s e n t i a l l y p r o p o s i t i o n 3.3 of

assertion is

~ 2 ] and the second assertion is theorem 3.5 of

[52].

Let us note t h a t t h i s theorem has a long h i s t o r y in the context of C * - a l g e b r a s (~5],

[18],

~ ] and ~ 7 ] ) .

1.13. Proposition {W.Krieger) (i)

:

The dimension group of an AF-groupoid is an E l l i o t

group and every E l l i o t t

133 group occurs as the dimension group of an AF groupoid. (ii)

Two AF-groupoids are isomorphic i f f

t h e i r dimension ranges are isomorphic.

The AF-groupoids considered in [52] have a compact u n i t space, but, as pointed out there, this assumption can be removed. Given an AF-groupoid G with compact u n i t space, the subgroup of i t s ample semi-group consisting of those G-maps which are everywhere defined is an ample CLF group in the sense of [52]. Conversely, given an ample CLF group acting on the space X, the groupoid o f the corresponding equivalence r e l a t i o n on X is AF. Let us describe the dimension range of the AF-groupoids that we have met in this section. The dimension group of the Glimm groupoid of the Ising model (and of the canonical anticommutation r e l a t i o n s ) is the group Q(2~) of r a t i o n a l numbers whose denominator is a power of

2, with the order i n h e r i t e d f r o m ~ . Its dimension range is the

segment [0,17 . With the notations of 1.6, the dimension of a cylinder set C(Zn) obtained by f i x i n g the spins inside a f i n i t e subset Zn of the l a t t i c e is D(C(Zn) ) = 2 _iZnI.Therel exists a unique p r o b a b i l i t y measure ~ on { - i , + 1 } z which extends D. I t is the unique ergodic i n v a r i a n t p r o b a b i l i t y measure of the groupoid. We give in the appendix a computation of the dimension group of the GICAR groupoid. I t is the group Z / I t ] of polynomials in one v a r i a b l e with integer c o e f f i c i e n t s , where the order is given by f > 0 i f f of a Riesz group (cf.

f(t)

> 0 f o r every t ~ ] 0 , 1 [ .

This is an example

[25]). There are uncountably many i n v a r i a n t ergodic p r o b a b i l i t y

measures, indexed by t c ] 0 , 1 l a n d obtained by composing the dimension map with the point evaluation at t . The measure corresponding to t = ~ 2 p r o b a b i l i t y measure for the CAR groupoid.

is the unique i n v a r i a n t

The dimension group of the skew-product o f the CAR groupoid and the number cocycle can be computed in the same fashion as the dimension group of the GICAR groupoid. I t is the group ] ~ ( t ) of r a t i o n a l functions with integer c o e f f i c i e n t s and whose only possible poles are at 0 and 1, where the order is given by f > 0 f(t)

iff

> 0 for every t ~ ] 0 , 1 [ . Let us look at the r e l a t i o n s h i p between AF-groupoids and AF C * - a l g e b r a s . I t is

due to Krieger ([52], theorem 4.1) and r e l i e s e s s e n t i a l l y on a r e s u l t of S t r ~ t i l ~

134

and

Voiculescu ( [ 6 9 ] ,

section I of chapter I ) . We give a s e l f - c o n t a i n e d proof which

is e s s e n t i a l l y the same as t h e i r s . Let us r e c a l l t h a t an AF C*-algebra is the induct i v e l i m i t o f a sequence o f f i n i t e - d i m e n s i o n a l C * - a l g e b r a s . Basic references f o r AF C ~ - a l g e b r a s are [ 3 ~ ,

[i~

and [ 8 ] .

The crux of the proof is the f o l l o w i n g lemma about f i n i t e - d i m e n s i o n a l C * - a l g e b r a s . 1.14. Lemma :

Let A be a f i n i t e - d i m e n s i o n a l

* - a l g e b r a and A 1 a s u b * - a l g e b r a . Then,

f o r any Cartan subalgebra B1 of A 1, there e x i s t s a Cartan subalgebra B of A which cont a i n s B1 and whose normalizer ~ ( B ) ,

t h a t i s , the inverse semi-group of p a r t i a l

isometries a o f A such t h a t d ( a ) , r ( a ) ~ B and a(Bd(a))a lizer

= B r ( a ) , contains the norma-

~N~(B1) of B1 in A I.

Proof :

Since A1 is a sum of simple , - a l g e b r a s , we may assume t h a t A1 i t s e l f

simple. The normalizer ~ ( B I )

o f B1 in AI contains m a t r i x u n i t s ( e i j )

i,j

is

= I ..... m

which span A1, The p r o j e c t i o n e l l of B1 decomposes in A i n t o minimal p r o j e c t i o n s : e11 = f l + . . . + f n . The f a m i l y ( e i l f j e l i )

i = 1 . . . . . m and j = 1 . . . . . n consists of

orthogonal p r o j e c t i o n s and is contained in a Cartan subalgebra B o f A. The algebra B1, which is spanned by the p r o j e c t i o n s ( e i i ) matrix units (eij)

i = 1 . . . . . m, is a subalgebra o f B. The

normalize B. Therefore Okrl(B1) is contained i n ~ ( B ) Q.E.D.

1.15. Proposition : (i) (ii)

Let A be a C * - a l g e b r a . The f o l l o w i n g p r o p e r t i e s are e q u i v a l e n t .

The C * - a l g e b r a A is AF. The C * - a l g e b r a A is the C * - a l g e b r a o f an AF-groupoid G. Moreover, under

these c o n d i t i o n s , the AF-groupoid G is unique up to isomorphism and i t s dimension range is the dimension range of A ( c f . Proof :

~7]).

Suppose t h a t A is an AF C * - a l g e b r a and choose an increasing sequence o f

f i n i t e - d i m e n s i o n a l C * - a l g e b r a s An which defines A. Construct by i n d u c t i o n a sequence o f Cartan subalgebras Bn of An such t h a t Bn+1 contains Bn and i t s normalizer J~rn+lin~+1 contains the normalizer J~rn o f Bn in An . Let B be the closure o f the union o f the I

Bn s. Since~N~ normalizes Bm f o r m ~ n, i t normalizes B, hence the ample inverse semigroup ~n o f Bn acts on B. We r e a l i z e

B as C * ( X ) ,

where X is a t o t a l l y

disconnected

135 l o c a l l y compact space and we l e t

~ = U ~ n , viewed as an inverse semi-group of p a r t i a l

homeomorphisms of X.The corresponding equivalence r e l a t i o n on X y i e l d s a p r i n c i p a l groupoid G which is AF because i t is of the form G = u Gn, where Gn is the p r i n c i p a l groupoid of the equivalence r e l a t i o n corresponding to ~n" I t is almost obvious t h a t Gn is an elementary groupoid. For,

~n p a r t i t i o n s

the atoms of the Boolean algebra Bn

of p r o j e c t i o n s of Bn i n t o equivalence classes. Let {YI,....,Ym}

be one of these classes

and l e t Y = Y l V . . . vY m. Then the reduction of Gn to Y is isomorphic to Y1 x Im, where I m is the t r a n s i t i v e

groupoid on m elements. The lemma allows the c o n s t r u c t i o n of

c o n s i s t e n t systems of matrix u n i t s in each algebra An . In other words, there e x i s t s a section k f o r the canonical map of J~r= uo~C n onto ~. Let C * ( ~n ) be the ( f i n i t e dimensional) sub C*-algebra of C * ( G ) generated by isomorphism #n of C * ( ~ n ) i n t o An such t h a t restriction

of #n+l to C * ( ~ n )

onto ~ An whose r e s t r i c t i o n

is

{Xs: s ~ ~n }. There e x i s t s an

#n(XS) = k(S) f o r S ~ ~n" Since the

0n, there e x i s t s an isomorphism # of W C* (~n)

to C* ( ~n ) is On . I t is isometric w i t h respect to the

C*-norms of C * (G) and of A, because f i n i t e - d i m e n s i o n a l * - a l g e b r a s have a unique C * - n o r m . Therefore, i t extends to an isomorphism of C*(G) onto A. The above argument also shows t h a t the C * - a l g e b r a of an AF-groupoid is AF. Let us keep the same notations as above. The dimension range D ( ~ n ) = ~ / ~ n is also the dimension range D(An) of the * - a l g e b r a An . The dimension range of G, which is the i n d u c t i v e l i m i t

of the dimension ranges D( ~n ), is equal to the dimension

range of the l o c a l l y f i n i t e

. - a l g e b r a uA n. I t is known (e.g.

~ 7 ] , remark 4.4,

page 34) t h a t t h i s is also the dimension range of A. Therefore, the uniqueness of the AF-groupoid G r e s u l t s from 1.13 ( i i ) . Q.E.D. 1.16. C o r o l l a r y :

Suppose t h a t a C * - a l g e b r a A has two Cartan subalgebras B1 and B2

which are both AF and which have countable l o c a l l y f i n i t e

ample semi-groups, then

B1 and B2 are conjugate by an automorphism of A, Proof :

The groupoids GI and G2 obtained by 2.4.15 are AF. (Therefore, the 2-cocycles

~i and ~2 are equal to 1). By the previous p r o p o s i t i o n , A

is AF and G1 and G2 have

the same dimension range. Therefore, they are isomorphic and an isomorphism of GI

136

onto G2 implements an automorphism o f A c a r r y i n g BI onto B2. Q.E.D. This is the only r e s u l t we have about the e x i s t e n c e and the uniqueness o f Cartan subalgebras. I t is not known, even in the case o f an AF C * - a l g e b r a ,

whether a

C ~ - a l g e b r a may have non-conjugate Caftan subalgebras. The f o l l o w i n g example shows t h a t the d e f i n i t i o n

we give o f a Cartan subalgebra

cannot be weakened i f we expect uniqueness. Let K be the a l g e b r a i c c l o s u r e o f a f i n i t e The m u l t i p l i c a t i v e

group o f K is denoted K ~ ,

field,

w i t h the d i s c r e t e t o p o l o g y .

i t s a d d i t i v e group is denoted K+ and

the dual group o f K+ is denoted K+. Since K is an i n c r e a s i n g sequence o f f i n i t e Kn, K+ is the i n d u c t i v e l i m i t finite

groups K+ n"

of finite

fields

groups K+n and K+ is the p r o j e c t i v e l i m i t

of

As a t o p o l o g i c a l space K+ is homeomorphic to the Cantor space.

The "ax + b" group over K is the s e m i - d i r e c t product G = K+ acts on K+ by m u l t i p l i c a t i o n .

x

K*, where K*

I t is equipped w i t h the product t o p o l o g y . We view K+ as

a normal a b e l i a n subgroup o f G. Since G has the d i s c r e t e t o p o l o g y , the C * - a l g e b r a B = C* (K +) is a subalgebra o f A = C ~ (G). 1.17. P r o p o s i t i o n : (i) (ii) (faithful)

Let A and B be as above.

The C ~ - a l g e b r a A is AF. The subalgebra B i s maximal a b e l i a n , r e g u l a r , is the image o f a unique c o n d i t i o n a l e x p e c t a t i o n but i t s ample semi-group does not a c t r e l a t i v e l y

f r e e l y on the spectrum K+ o f B, hence i t

fails

to be a Cartan subalgebra.

Proof : ( c f . Dixmier [ 1 7 ] ) . (i)

As above, we w r i t e K as union o f an i n c r e a s i n g sequence o f f i n i t e

fields

Kn-

The "ax + b" group over Kn, Gn, is a subgroup o f G and G i s the union o f the Gn'S. As in 1.15, we see t h a t C * ( G )

is the i n d u c t i v e l i m i t

o f the C*(Gn)'S, which are

finite-dimensional. (ii)

As an i n c r e a s i n g union o f f i n i t e

groups, G i s amenable. We apply 2 . 4 . 2 , to

view the elements o f C*(G) as f u n c t i o n s on G vanishing a t i n f i n i t y .

The elements o f

I37 C * ( K +) are those f u n c t i o n s which vanish outside K+. To show t h a t B is maximal a b e l i a n , we pick an element f of i t s commutant in A. I t s a t i s f i e s ~ b l . f ~ e _ b I = f f o r every b I a K+, where Cbl is the p o i n t mass at b I . E x p l i c i t l y , f ( a , ( 1 - a ) b I +b) = f ( a , b ) nity,

t h i s gives

f o r every b I ~ K+, a E K*, b ~ K+. Since f vanishes at i n f i -

t h i s is only possible i f f ( a , b ) = 0 when a # 1, t h a t i s , f ~ B. Since K+ is a normal subgroup, the normalizer of B contains the elements ex'

where x c G. Therefore B is r e g u l a r . Let P be a c o n d i t i o n a l expectation onto B. From the r e l a t i o n s (l,bl)

(a,b) ( l , b l ) - 1

(1,bl)

(a,b) = (a,b + bm)

f o r every a s K* P(E(a,(l-a)bl translation. restriction

= ( a , ( 1 - a ) b I +b) and

and every b,b I ~ K+, we obtain t h a t P(a(a,b)) =

+b) = C ( l - a ) b l * P(a(a,b) )" Thus, i f a # 1, P(a(a,b)) Since i t vanishes at i n f i n i t y ,

of P to Cc(G ) is the r e s t r i c t i o n

is i n v a r i a n t under

i t must be zero. This shows t h a t the map of Cc(G ) onto Cc(K+). On the other

hand, i t r e s u l t s from 2.2.9 t h a t t h i s r e s t r i c t i o n

map is p o s i t i v e and bounded. Hence

i t extends uniquely to a c o n d i t i o n a l expectation of C * ( G ) onto C * ( K + ) , which is still

given by r e s t r i c t i n g

a f u n c t i o n to K+. I t is c l e a r l y f a i t h f u l .

To show t h a t the ample semi-group of B does not act r e l a t i v e l y

f r e e l y on K+,

we note t h a t the element C(a,b) of the normalizer of B induces the homeomorphism s a of i t s spectrum K+, where Sa(× ) = ax

and a×(b) = x(ab) f o r × ~ K+. I f a # I , the

set of f i x e d points of s a is reduced to the i d e n t i t y character 1, hence is not open in K+ . Q. E.D. We have not been able to determine whether the exact sequence "~+~(B)

÷~(B) # ~(B) +%

s p l i t s or not. 1.18. Remark : The C * - a l g e b r a A is the C * - a l g e b r a of the transformation group (K+,K*) where the a c t i o n of K*

on K+ is described above. Since Y = K + \ { I } is an

i n v a r i a n t open subset of K+, A is an extension of C* (Y x K*) by C ~ ( K * ) One can show t h a t the dimension range of the groupoid Y × K*

(2.4.4).

is the segment [O,p[

138 of the dimension group Q(p®) of r a t i o n a l numbers whose denominator is a power of p, where p is the c h a r a c t e r i s t i c of the f i e l d

K. Therefore, C * ( Y × K*) is a matroid

algebra without u n i t of type Mp~ p ( n o t a t i o n of [18]). On the other hand, the C * - a l g e b r a C*(K*) is the C * - a l g e b r a of the Cantor space. I t r e s u l t s from [25], section 5.1, t h a t the dimension group of A is an extension of Q(p~) by the dimension group o f the Cantor space. Let us mention here, w i t h o u t g i v i n g the d e t a i l s , t h a t such extensions are c h a r a c t e r i z e d , up to equivalence, by measures on

the Cantor space. E x p l i c i t l y ,

f i n d s t h a t Ko(A ) = Q(p~) x C ( K * , Z ) and t h a t an element ( q , f ) only i f q +

one

is p o s i t i v e i f and

# ( f ) is p o s i t i v e where # is the measure on K*, constructed as f o l l o w s .

Let ( n i ) be a sequence o f integers such t h a t n i d i v i d e s ni+ 1 and n I = I , l e t ni qi = p and l e t f i

-

qi - 1

f o r i > 2 and f l

= p" Realize the space K ' a s

the

qi -1-1 product space ~ {1,2 . . . . . f i }. The measure # is concentrated on the points ( a i ) i=1 with a i = 1 f o r i large enough. I f k is the l a s t index i f o r which a i # 1 (or, i f a I = i f o r every i •

set k = I) ,

the measure of the p o i n t ( a i ) is

P j=k qj-1

,

i i _ i j. !~ J qj+ll L

The dimension range of A is the segment [O,c], where c is the element (p-1,1) of

Q(p~) × c(~*, ~ . Another method to check t h a t the subalgebra B is not a Cartan subalgebra is to determine i t s dimension range and i t s dimension group r e l a t i v e to A. Its dimension group is an extension of Q(p~) by Z .

2.

The Groupoids On

The aim o f t h i s section is to e x h i b i t the C * - a l g e b r a s generated by isometries introduced by J.Cuntz in [ l ~ a s

the C * - a l g e b r a s o f a groupoid. The groupoids we

construct are not p r i n c i p a l and we do not know i f these algebras can be r e a l i z e d as the C * - a l g e b r a s o f a p r i n c i p a l groupoid. Nevertheless, t h i s d e s c r i p t i o n o f the Cuntz algebras reveals much o f t h e i r s t r u c t u r e .

I t also makes apparent the r e l a t i o n s h i p

between these algebras and some inverse semi-groups.

139

We s t a r t w i t h a crossed product c o n s t r u c t i o n prompted by the r e p r e s e n t a t i o n o f the Cuntz algebras as a crossed product ( s e c t i o n 2 o f n = I , which w i l l

give the algebra o f the b i c y c l i c

[ 1 5 ] ) . We include the case

semi-group, s t u d i e d by Barnes in

[1]. For every n = 1,2 . . . . . ~, we d e f i n e the f o l l o w i n g AF-groupoids Gn. The groupoid G1 is the compact space Z = Z u { ~ } ,

o n e - p o i n t c o m p a c t i f i c a t i o n o f the space o f i n t e g e r s

w i t h i t s d i s c r e t e t o o o l o g y . We r e c a l l lation u ~ v iff

corresponds to the equivalence r e -

u = v on ~ .

For n l a r g e r than 1 . b u t f i n i t e , relation u ~ v iff {0,1 . . . . . n - l }

that it

ui = vi for all

the groupoid Gn corresponds to the equivalence but f i n i t e l y

many i ' s

on the compact space

w i t h the product t o p o l o g y . This i s a Glimm groupoid. I t s dimension

group is the group ~(n ~) o f r a t i o n a l numbers whose denominator i s a power o f n, w i t h the o r d e r i n h e r i t e d from ~ and i t s dimension range is the segment [0,1] ~. _z The u n i t space o f the groupoid G i s the space G0~ = {u e ~ : u i = 0 f o r i sufficiently

small and uj = ~ f o r every j > i i f u i = ~ } , where ~-~= N u { ~ } . The

c y l i n d e r sets Z ( ~ ) , where ~ = ( . . . . O , j k . . . . . jk+L) w i t h k e ~ , L e~q and Jk+i c ~ , and t h e i r complements form a subbase o f open sets f o r a t o p o l o g y on G~. This t o p o l o g y i s l o c a l l y compact and t o t a l l y

disconnected. The

c y l i n d e r sets Z(m

d e f i n e , f o r u ~ G~, k(u) as the s m a l l e s t index i such t h a t u i as ~ i f u i < ~ f o r every i . on G~

: u ~ v iff

The groupoid G

are compact. We

~, i f

i t e x i s t s and

corresponds to the equlvalence r e l a t i o n

k(u) = k(v) and u i = v i f o r a l l

as in the example of the Glimm groupoids t h a t i t o r b i t o f a p o i n t u i s [u] = {v : k(v) ~ k ( u ) } .

but f i n i t e l y

many i ' s .

One checks

is an AF-groupoid. The c l o s u r e o f the

In p a r t i c u l a r ,

t h e r e are dense o r b i t s .

The i n v a r i a n t open sets form a decreasing sequence ( U i ) , i m ~ , where Ui = {u : k(u) > i } . sum i B y ( c f .

The dimension group o f the groupoid G

is the l e x i c o g r a p h i c a l d i r e c t

5.3 o f [ 2 8 ] ) and i t s dimension range is the whole p o s i t i v e cone. F u r t h e r

references to the

AF C * - a l g e b r a s whose dimension group i s t o t a l l y

ordered can be

found in [28]. In each case, there e x i s t s a n a t u r a l s h i f t normalizes the ample semi-group o f Gn, t h a t i s ,

#0 on the u n i t space of Gn which such t h a t f o r any G-map s in the ample

semi-group o f Gn, #0 o s o #0-1 i s also in the ample semi-group o f Gn. E x p l i c i t l y ,

140

for

n = i,

the s h i f t

~0 one_ sends u i n t o

u - 1 if

u is finite

n : 2, ~0 i s g i v e n by ¢Ou = v where v i = ui_ 1. The s h i f t o f the l o c a l l y We l e t

Z

compact g r o u p o i d Gn. E x p l i c i t l y ,

t i n g measures on the f i b e r s Finally,

we d e f i n e

It

¢0 induces an automorphism

# i s g i v e n by ~ ( u , v )

a c t on Gn by z + ~z and form the s e m i - d i r e c t

the b e g i n n i n g o f the p r o o f o f 2 . 3 . 9 ) .

and ~ i n t o ~. For

= (~O(u),~O(v)).

p r o d u c t Gn x ~ ( s e e

i s an r - d i s c r e t e

1 . 1 . 7 and

groupoid admitting

the coun-

as Haar system.

f o r each n the f o l l o w i n g

n = I,

O~ = ~ = I N u { ~ } .

It will

be i d e n t i f i e d

with

{0,1 ..... n-l} ~

It will

be i d e n t i f i e d

with

{u s ~

subset of unit

For 2 _< n < ~, OOn = {u ~ { 0 , 1 , . . . , n - I }

space o f Gn. For

~

: ui = 0 for

i < 0}.

For n = ~, 00 = {u s GO : u i = 0 f o r

: ui = ~ ~

uj = ~ f o r e v e r y j ~ i } .

i < 0).

Each o f

t h e s e subsets 00 i s c l o s e d in GO hence i s a compact space n n' " 2.1.

Definition

semi-direct

:

L e t n = 1,2 . . . . . ~. The Cuntz g r o u p o i d On i s the r e d u c t i o n

p r o d u c t Gn × # ~ t o

the unit

01 = { ( u , z )

of

of its

unit

of the

space ( i d e n t i f i e d

with

space o f Gn).

L e t us s p e l l

(u,z)(u

0 the c l o s e d s u b s e t On

o u t the a l g e b r a i c

~ ~-]xZ

+ z, z')

(u,z)

: u + z ~},

= (u,z + z')

i s u and i t s

structure

o f the g r o u p o i d s On . For n = 1,

where ~ + z . . . .

The m u l t i p l i c a t i o n

and the i n v e r s e o f ( u , z )

domain u n i t

is

i s u + z. For n g r e a t e r

i s g i v e n by

(u + z , - z ) .

The range u n i t

than 1 but f i n i t e ,

N

On = ~ ( u , v , z ) ui = vi_ z

e {0,1 .....

for

The m u l t i p l i c a t i o n is (v,u,-z).

0

but f i n i t e l y

The range u n i t

= k(v)

The m u l t i p l i c a t i o n

× {0,1 .....

of (u,v,z)

e 00 × 00 × Z :

n-l}

×~:

many i ' s ) .

i s g i v e n by ( u , v , z )

= {(u,v,z)

k(u)

all

n-l}

(v,w,z')

= (u,w,z + z')and

i s u and i t s

domain u n i t

ui = vi_ z for all

= z i n the case when k(u) o r k ( v )

the i n v e r s e o f ( u , v , z )

i s v.

but f i n i t e l y

In the case

many i ' s

and

is finite}.

and the i n v e r s e are g i v e n as a b o v e .

The n e x t t a s k i s t o d e t e r m i n e t h e ample s e m i - g r o u p o f t h e g r o u p o i d s On . L e t us first

define

section of

t h e Cuntz i n v e r s e s e m i - g r o u p [1~.

On , introduced

The s e m i - g r o u p 01 i s the b i c y c l i c

implicitly

semi-group

[11].

in the first

141

2.2.

Definition

:

L e t n = 1,2 . . . . . ~.

semi-group consisting letters qiPi

Pi'

o f an i d e n t i t y

1, a zero e l e m e n t 0 and a l l

qi w i t h i = I . . . . . n, s u b j e c t

to the relations

qjPi

On

i s the

the words i n the = 0 if

i # j and

= 1. L e t us r e c a l l

the n o t a t i o n s

let

Wkn = {~ = ( J l ' " " J k )

For

= ( J l . . . . . Jk )

it

The Cuntz i n v e r s e s e m i - g r o u p

i s shown i n

[15]

~

: Ji W~, l e t

and 0 "1 = O. I t s 0 o r d e r on O n i s

~

[15].

n W0 =

{0,1 ..... n-l}},

p~

=

PJIPJ2

.

"'PJk

i k) 2 ( J l . . . . . j ~ )

The compact open G-sets o f Gn x c Z compact open G-set o f Gn and z m Z . : u e S}

n { 0 } and Wn~ = k=OU Wk-

and q~

= qJkqJk-1"

iff

O 0n = {pmqm :m

where z c Z

(p~qB) - I W~}

k Z L a n d i m = Jm f o r

a r e o f t h e form S ×

Therefore,

..qj

Then,

1"

may be u n i q u e l y w r i t t e n

O n i s an i n v e r s e s e m i - g r o u p w i t h

set of idempotent elements is (i I .....

the form { ( u , z )

L e t n = 1,2 . . . . . ~. Given k c ~ ,

(lemma 1.3) t h a t any word i n p i q i

W~n . The s e m i - g r o u p

w i t h ~,~ e

of

pmq~

= pBqm, 1-1=1

{0,1}.

The

m = 1.....

L.

{ z } where S i s a

the compact open G-sets o f On are o f

and S i s a compact open subset o f ~4 such t h a t

S + z c N i n t h e case n = 1 and o f t h e form

{(u,v,z)

E 00 x 00 x Z : n n

(u,v(-z))

e S}

where S i s a compact open G - s e t o f Gn and [ v ( - z ) ] i = v i _ z , i n the case n > 2. In particular,

let

us d e f i n e ,

f o r e v e r y n = 1 , 2 , . . . ,~ and e v e r y m,B ~ W~n t h e f o l l o w i n g

compact open G-sets o f On . For n = 1, S(m,B) = { ( u , ~ ( ~ ) where t h e l e n g t h

L(m) o f m i s k i f

{((~,u),(B,u),~(~) 2.3.

- ~(B))

Proposition

and 1 i n t o O0n

:

- L(m))

: u e [~(~),~]},

m i s i n W~. For n > 2, S(m,~) =

: u ~ o~).

L e t n = 1,2 . . . . . =. The map which sends P~qB i n t o

i s an isomorphism o f the i n v e r s e

o f t h e g r o u p o i d On . I t s

image, which w i l l

s e m i - g r o u p (On i n t o

S(~,~),

0 into

the ample s e m i - g r o u p

a l s o be denoted On, g e n e r a t e s the ample

s e m i - g r o u p i n t h e sense t h a t (i)

e v e r y compact open s e t o f 00 may be w r i t t e n n

two sets A and B which are both a f i n i t e (ii)

union of elements of

e v e r y compact open G-set o f On may be w r i t t e n

where ( E i ) and ( F i ) Si ' s

disjoint

are in

On .

a r e two f a m i l i e s

of disjoint

as the d i f f e r e n c e

as a f i n i t e

A\B

of

0 0n.

union u EiSiFi 1 compact open sets i n O0 n and t h e

142 Proof : it

The map i s c l e a r l y o n e - t o - o n e . In o r d e r to show t h a t i t

s u f f i c e s to c o n s i d e r the generators Pi and q i ' s .

W~,P

i s a homomorphism,

Let us d e f i n e , f o r m and B in

= S(~,O) and QB = S(O,B). Thus, S(~,F) = P Q~. We w r i t e Pi i n s t e a d o f P ( i ) .

Then, the f o l l o w i n g r e l a t i o n s are s a t i s f i e d and QiQj = Q ( j , i ) "

: QjPi = @ i f

i # j,

Therefore, the map is an isomorphism o f

OiP i = 0 0n ' •

PiPj=P(

i,j)

O n i n t o the ample semi-

group o f On . The image o f the idempotent element pmqm is the i n t e r v a l

[~(m),~]

in the case

n = 1 and the c y l i n d e r set Z(~) o f 0 0 in the case n > 2. In the case n - I , the n

assertion (i)

is clear.

I t i s also c l e a r in the case 2 E n < ~

open s e t o f { 0 , I . . . . . n - l } m the case n = =, i t

is a finite

disjoint

since every compact

union o f c y l i n d e r sets Z(~).

In

s u f f i c e s to check t h a t the sets A\.B, where both A and B are unions

o f c y l i n d e r sets Z(m) form a base f o r the t o p o l o g y o f 00. This is immediate from the definition

o f the t o p o l o g y o f 00.

The l a s t a s s e r t i o n is also c l e a r . For example, in the case n > 2, the G-set {(u,v,O)

: ( u , v ) ~ S} where S corresponds to the t r a n s p o s i t i o n (~,u) ÷ ( ~ , u ) , w i t h

m,B e Wn belongs to k'

0 n. 0 E.D.

2.4. Remark : The groupoid On has the p r o p e r t y o f having i t s ample semi-group generated by the inverse semi-group O n . Two questions a r i s e f o r which we have no answer. Given an inverse s e m i - g r o u p ~ , does there e x i s t an r - d i s c r e t e groupoid G whose ample semi-group o f compact open G-sets i s generated by ~

and covers G ?

What kind o f uniqueness can we expect ? Let us note t h a t the r e a l i z a t i o n

o f an inverse semi-group

o f G-sets i n t r o d u c e s an e x t r a s t r u c t u r e on ~ and embeds i t s i n t o a Boolean algebra and a l l o w s the d e f i n i t i o n be added provided t h a t r(S)

n

of a partial

~ as a semi-group

idempotent elements a d d i t i o n : S and T can

r(T) = ~ and d(S) n d(T) = ~, then S + T i s the union

o f S and T. For example, we have introduced in the case 2 < n < ~ the r e l a t i o n n

i=1

PiQi = I .

2.5. P r o p o s i t i o n : (i)

For every n = 1,2 . . . . . ~, the groupoid On is amenable.

143

(ii)

For n = 1, j ~ i s

the t r a n s i t i v e (iii) (iv)

an open i n v a r i a n t set f o r 01 . The r e d u c t i o n o f 01 t o

groupoid on ~

~

is

and the r e d u c t i o n o f 01 to {~} is the group Z -

For n > 2, the groupoid On i s m i n i m a l . For every n = 1,2 . . . . . ~, the groupoid On has a base o f compact open G-sets

and i s o f i n f i n i t e

type.

Proof : (i)

We have c o n s t r u c t e d On as the r e d u c t i o n o f a s e m i - d i r e c t p r o d u c t . L.~emay

apply 2 . 3 . 7 and 2 . 3 . 9 . (ii)

The open subset o f IN o f IN i s c l e a r l y

isomorphism o f 011 ~ o f 01 a t {~} (iii)

We may d e f i n e the

onto IN×IN which sends ( u , z ) i n t o (u,u + z ) . The i s o t r o p y group

is i.

The groupoid On induces the equivalence r e l a t i o n ~ on i t s u n i t space,

where u ~ v i f f i's.

invariant.

there e x i s t s z ~ s u c h

t h a t u i = vi_ z f o r a l l

but f i n i t e l y

many

Hence every o r b i t meets every c y l i n d e r set Z ( ~ ) , where ~ ~ Wn f o r 2 < n <

and every c y l i n d e r set Z ( ~ Z ( ~ j ) , orbit

where ~,Bj ~ W~for n =

~,

~. This shows t h a t every

i s dense.

(iv)

The G-sets SE, where S c O n and E is a compact ooen set in OOn c o n s t i t u t e

a base f o r the t o p o l o g y o f On . Since, in On i s o f i n f i n i t e

O n , p i q i i s e q u i v a l e n t to 1, the groupoid

type. Q.E.D.

Let us r e c a l l

the d e f i n i t i o n

o f a r e p r e s e n t a t i o n o f an inverse semi-group on a

H i l b e r t space given by B. Barnes in [ 1 ] , page 363. 2.6. D e f i n i t i o n

:

A r e p r e s e n t a t i o n o f an inverse semi-group ~ o n a H i l b e r t space H

is an inverse semi-group homomorphism o f ~ i n t o

an inverse semi-group o f p a r t i a l

i s o m e t r i e s o f H. Let V be a r e p r e s e n t a t i o n o f the inverse semi-group On , n = 1,2 . . . . . ~. The images Si = V(Pi) o f the generators Pi are i s o m e t r i e s w i t h m u t u a l l y orthogonal ranges. Conversely, any sequence ( S i ) i = i , . . . . n o f i s o m e t r i e s w i t h m u t u a l l y orthogonal ranges d e f i n e s a unique r e p r e s e n t a t i o n V o f

O n such t h a t V(Pi) = Si f o r every i = 1 . . . . . n.

144

n

In

the case 2 _< n

< ~, we r e q u i r e t h a t

~ SiS i = ] . i=1

2.7. P r o p o s i t i o n :

L e t n = 1,2 . . . . . ~. There i s a b i j e c t i v e

the r e p r e s e n t a t i o n s

V of

correspondence between n

i n t h e case 2 < n Proof :

O n on s e p a r a b l e H i l b e r t s p a c e s , such t h a t

~ V(Pi)V(qi) = 1 1

< ~, and t h e r e p r e s e n t a t i o n s o f C * ( O n ) on s e p a r a b l e H i l b e r t

Suppose t h a t L i s a r e p r e s e n t a t i o n o f C *(On)

Then, by 2 . 1 . 20, i t

spaces.

gives

by r e s t r i c t i o n

a r e p r e s e n t a t i o n o f the ample semi-group o f On , hence a r e p r e s e n t a t i o n n o f O n . In t h e case 2 5 n < ~, the r e l a t i o n ~ Ei = I , where Ei is the c h a r a c t e r i s i=1 t i c f u n c t i o n o f the c y l i n d e r s e t Z ( i ) , holds in C * (On) and g i v e s the r e l a t i o n n

Z sis = i

i=1

C o n v e r s e l y , s u p p o s e \ t h a t V is a r e p r e s e n t a t i o n o f On such t h a t , n

2 ~ n < =,

ZI

in the case

0

S i S i * = 1, where Si = V ( P i ) . I t s r e s t r i c t i o n to the set O n o f idempotent i= e l e m e n t s , which w i l l be denoted M, is a monotone p r o j e c t i o n - v a l u e d f u n c t i o n , t a k i n g t h e v a l u e 0 a t 0 and the v a l u e i a t I • I t

is finitely

additive

in the case 2 s n < ~,

n

because o f t h e r e l a t i o n jection-valued

i ~= 1 S i S i* = I . We w i l l

extend it

additive

pro-

measure on the Boolean a l g e b r a ~ n o f compact open subsets o f 00. In n

t h e case n = 1, any compact open subset o f ~ i s o f elements o f

O . Thus, i f

M(A) = 1~=1=M(B i ) M is f i n i t e l y

to a finitely

a finite

disjoint

union o f d i f f e r e n c e

u Bi \C i w i t h T i , Ci e O and Ci c Bi , i=l - M ( C i ) . Because the o r d e r o f ~ i s t o t a l , M(A) is w e l l

additive.

A =

we d e f i n e d e f i n e d and

In the case 2 ~ n < ~,any compact open subset o f

~N

{0,i ..... n-l}

is a f i n i t e

disjoint

union o f elements o f

O 0 Thus, i f A = u B i , n" i=1 M ( B i ) . This is w e l l d e f i n e d and a d d i t i v e because

0 w i t h Bi ~ On, we d e f i n e M(A) = ~ 0 i=I M is a d d i t i v e on O n . In t h e case n = ~, we f i r s t are a f i n i t e

disjoint

union o f elements o f

extend M to the elements o f

0

which

0 O n . Since e v e r y element A o f O~ i s the

d i f f e r e n c e o f two such e l e m e n t s , say A = B\C w i t h C c B, we may d e f i n e M(A) = M(B)-M(C). One shows as i n the case n = 1 t h a t M i s w e l l

d e f i n e d and a d d i t i v e .

Having extended M t o the Boolean a l g e b r a ~ n ' o f the ample semi-group o f On . We know from 2 . 3 . ( i i i ) o f On may be w r i t t e n families of disjoint

as a f i n i t e

we may extend V t o a r e p r e s e n t a t i o n t h a t e v e r y compact open G-set

union S = u EiSiF i where ( E i ) and ( F i ) are two 1 elements o f ~ n and the Si ~S are in O n . We d e f i n e

145 g V(S) = ~ M(Ei)V(Si)M(Fi). I t is a p a r t i a l isometry and i t does not depend on the I way S has been w r i t t e n . Moreover, i t is an inverse semi-group homomorphism. The pair (V,M) is a covariant representation of On (cf.2.1.20) and can be extended to a representation of C*(On). E x p l i c i t l y , every f c Cc(On) may be written f =

Z1 hixsi where hi a C c (00 n ) and Si is a compact open G-set of On . We define L(f) =

M(hi)V(Si). A computation s i m i l a r to one given in the proof of 2.4.15 shows that I L(f) is well defined. Moreover, the map L so defined is a representation of Cc(On) continuous for the inductive l i m i t topology. Since r - d i s c r e t e groupoids with Haar system have s u f f i c i e n t l y many non-singular Borel G-sets, we know from 2.1.22 that L extends to a representation of C* (On). Q.E,D. 2.8. Remarks : ( i ) In order to study the representations of an inverse semi-group ~on a H i l b e r t space, B. Barnes makes use in [1]

and [2]

of i t s Banach .-algebra 1 ( ~ ) .

He shows in p a r t i c u l a r that LI(~) has a f a i t h f u l representation. The example of On suggests another approach. One can t r y f i r s t to realize the inverse semi-group as a generating subsemi-group of the ample semi-group of a groupoid G and then define the C* -algebra o f ~ as C*(G).

The example of the b i c y c l i c semi-group 01 is studied

in [1] (section 7). The description of i t s irreducible representations given there can also be obtained from 2.5. ( i i ) . (ii)

The C*-algebra C *(On) is generated by the isometries Pi' i = i . . . . . n.

Indeed these isometries generate OnaS an inverse semi-group. Moreover O~ generates the Boolean algebra ~n of compact open subsets of OOn" Therefore, the C*-algebra generated by the Pi's contains Cc(On). I t must be C* (On). Thus C*(On) i s , for n ~ 2, one of the C*-algebras studied by J. Cuntz in [15]. I t is shown there that such an algebra is simple. We can prove i t d i r e c t l y . Indeed the groupoid On is amenable, minimal and e s s e n t i a l l y principal ( d e f i n i t i o n 2.4.3). Hence we may apply proposition 2.4.6. We have seen (1.1.7) that the semi-direct product Gn ×¢2 has a natural cocycle

146 c n c ZI(Gn × ¢ Z , Z ) , reduction Gn x¢ ~

namely the cocycle given by Cn(X,Z ) = z. I t s r e s t r i c t i o n iO~ is s t i l l

f o r n=l, c I ~ Z I ( o I , Z )

a cocycle. E x p l i c i t l y ,

to the is

defined by Cl(U,Z ) = z and f o r n > 2, c n c z l ( O n , ~ ) is defined by Cn(U,V,Z ) = z. We may observe t h a t the " f i x e d p o i n t " groupoid Gn. Indeed, f o r n = 1, c n l ( O ) i s

cnl(O) bears some resemblance w i t h

the u n i t s p a c e ' o f

01 . For 2 < n < ~, c n l ( O ) i s

Glimm groupoid given by the equivalence r e l a t i o n u ~ v i f f finitely

many i ' s on {0,1 . . . . . n - l }

For n =

by the equivalence r e l a t i o n u ~ v i f f

the

u i = v i f o r a l l but

~, c~1(0) is the AF groupoid given

k(u) = k(v) and u i = v i f o r a l l but f i n i t e l y

many i ' s on 00. Its dimension group is the l e x i c o g r a p h i c a l d i r e c t sum

77 and

i~IN i t s dimension range is the segment [ 0 , i ] ,

where 1 = (1,0,0 . . . . ).

The f o l l o w i n g r e s u l t , due to Olesen and Pedersen ( [ 5 8 ] ) ,

is i n t e r e s t i n g because

i t e x h i b i t s the d i f f e r e n t behavior o f the On groupoids, in comparison to the AF groupoids, w i t h respect to KMS measures. The d e f i n i t i o n o f (c,6) KMS measures has been given in 1.3.15. We w i l l make use o f the r e l a t i o n d u ' s - I (u) = D ' l ( u s ) , where D is the modular d~ f u n c t i o n of u and s is a G-map, established in 1.3.18. ( i i i ) and 1.3.20. 2.9. P r o p o s i t i o n : (i)

Let n = 1,2 . . . . . ~

and l e t c n c Z1(On,~) be as above. Then

i f n = 1, there are no (cI,B)-KMS p r o b a b i l i t y measures f o r 6 > 0 and there

e x i s t s a unique (Cl,~)-KMS p r o b a b i l i t y measure f o r ~ < 0 ; (ii)

i f n > 2, there are no (Cn,~)-KMS p r o b a b i l i t y measures f o r ~ # logn and

there e x i s t s a unique (Cn,6)-KMS p r o b a b i l i t y measure f o r ~ = logn. Proof : (i)

Since d - l ( u ) = { ( v , u - v )

d-l(~) = {(~,z) triction

: z E Z},

of c I to d - l ( u )

such t h a t the r e s t r i c t i o n

: v ~},

i f u is f i n i t e

Minlcl), which is the set of u n i t s u such t h a t the resis non-negative, is empty, while the set MaXlCl)Of u n i t s u of c I to d - l ( u )

is non p o s i t i v e is { 0 } . Therefore there is

no ~ KMS measure and the p o i n t mass at 0 is the unique -~ Suppose t h a t ~ is a KMS p r o b a b i l i t y measure on N a t G-set { ( u , l )

: u ~}and

subset A o f ~

and

KMS p r o b a b i l i t y measure. a finite

~. Let Q be the

l e t q be the corresponding G-map. For every compact open

, the f o l l o w i n g e q u a l i t i e s hold :

147

u(A-q) = ~×A(U) d ( ~ . q - 1 ) ( u ) = ~XA(U) D - l ( u q ) du(u) = e~

In p a r t i c u l a r ,

~

(A)

.

f o r every i E N ,

~{i + i} = e~

u{i}.

Since ~ i s r e q u i r e d t o be a

probability

measure, t h i s i s p o s s i b l e f o r B ~ 0 o n l y . Then u i s u n i q u e l y d e f i n e d by e Bi - - i f B < 0 and by ~ {~} = i f o r B = O. 1 - e8

p{i}

(ii) {(v,u,z)

Suppose 2 ~ n < ~. Then M i n ~ n )= Max(Cnl= 0 because d - l ( u ) : v ~ u and z E 7 }

probability

and t h e r e are no KMS measures a t i n f i n i t y .

measure on 0 0 a t a f i n i t e

~

We know ( 1 . 3 . 1 6 )

however the Glimm g r o u p o i d c n l ( O ) has a unique i n v a r i a n t o f the s t r u c t u r e

of its

probability

invariant

measure, because

~n d e f i n e d by

un(Z(m)) = n -~(m) f o r every m c W~n" Since ~n(A.pmq~) = n(~(m) (~,B)

Let ~ be a KMS

that u is cnl(O)

dimension range. This i s the measure

f o r every compact open s e t A and every p a i r

=

- ~(B)) un(A )

in W~n' the modular f u n c t i o n

w i t h r e s p e c t t o 0 ~n i s Dn(U,V,Z ) = n -z = e x p ( - l o g n c ( u , v , z ) )

o f ~n

Thus ~ must be equal t o

l o g n. (iii) and d - l ( u )

Suppose n = ~. Since d - l ( u ) = {(v,u,z)

: v ~ u, z c ~ }

{(v,u,k(v) if

- k(u))

k(u) i s i n f i n i t e ,

and Min(c ) = {~} where = denotes the sequence ( ~ , ~ , measures a t ~ = -~

which i s the o n l y p r o b a b i l i t y

invariant

k(u) i s f i n i t e

Max(c I i s always empty Thus t h e r e i s no }<MS

and the p o i n t mass a t { = } , ~ , i s the o n l y KMS p r o b a b i l i t y

measure a t B = ~. There cannot be any KMS p r o b a b i l i t y ~,

...).

: v ~ u} i f

measure i n v a r i a n t

measures a t a f i n i t e under c ~ l ( o ) ,

~ because

i s not q u a s i -

under 0 . Q.E.D.

2.10.

Remark :

subgroup o f ~ , are i n v a r i a n t

I f G i s the g r o u p o i d o f a t r a n s f o r m a t i o n the o n l y p o s s i b l e KMS p r o b a b i l i t y probability

measures.

group ( U , S ) , where S i s a

measures f o r the c o c y c l e c ( u , s )

= s

•

Appendix. THE DIMENSION GROUP OF THE GIDAR ALGEBRA

We have seen t h a t AF groupoids are c l a s s i f i e d ii).

by t h e i r dimension ranges ( 3 . 1 . 1 3 .

T h e r e f o r e , the computation of the dimension range is the e s s e n t i a l step in the

study o f AF groupoids. The problem can be s p l i t

i n t o two p a r t s , f i r s t

the computation

o f the dimension group, second the d e t e r m i n a t i o n of the dimension range as an upward d i r e c t e d h e r e d i t a r y subset o f the p o s i t i v e cone o f the dimension group. The f i r s t i s more d i f f i c u l t .

An E l l i o t t

group, t h a t i s , the dimension group o f an AF g r o u p o i d ,

i s u s u a l l y given as an i n d u c t i v e l i m i t .

Although some i n f o r m a t i o n can be read o f f

from the corresponding diagram, i n p a r t i c u l a r

its

i d e a l s t r u c t u r e (see [ 8 ] ) ,

can decide when two diagrams g i v e isomorphic dimension groups, i t r e s t to have an i n t r i n s i c

part

definition

and one

is o f g r e a t i n t e -

o f the dimension group. This is why t h i s compu-

t a t i o n is included here, which is probably known to o t h e r s .

Let us r e c a l l CAR = { ( u , v )

the d e f i n i t i o n s

: the CAR groupoid is

~ U x U : u i = v i a . e . } where U = { 0 , 1 } ~ the GICAR groupoid is the

subgroupoid GICAR = c - l ( o )

where c ( u , v ) = ~u i - v i .

I t s ample semi-group c o n s i s t s o f

the G-maps o f CAR which "preserve the number o f p a r t i c l e s " . f o r j l a r g e enough. W r i t i n g GICAR as an i n d u c t i v e l i m i t ,

we o b t a i n the f o l l o w i n g

i n d u c t i v e system f o r i t s dimension group :

....

~n

li il

,~n+l

___, . . .

J

This means # ( u . s ) i

n = 1,2 . . . .

J

= # ui

149

Proposition

:

The dimension group o f GICAR i s ~ [ t ]

with integer coefficients} t

w i t h usual a d d i t i o n

= { p o l y n o m i a l s in

and o r d e r

f

> 0 iff

f(t)>

t 0 f o r any

a ]O,I[.

Proof : (a) We f i r s t (1) t p

observe t h a t f o r a g i v e n n and p = 0 . . . . . n, ~ ( n - p~-~ t k k-D/ (I k=O

:

11

t)n-k

-

and

L

k=0

(b) ~le i n t r o d u c e e kn =

t k (1 - t ) n - k

f o r n = 1,2 . . . .

and k = 0 . . . . .

n

and remark t h a t (2) the e nk ' s

generate ~ [t],

(3) f o r f i x e d n, the e n' k s

are l i n e a r l y

n n+l n+l (4) e k = e k + ek+ 1 n = 1,2 . . . . Hence, as a g r o u p ~ [ t ]

and k = 0,1 . . . . . n.

i s the l i m i t

o f the i n d u c t i v e

the o r d e r the system induces o n , I t ] . l a r g e n, the c o e f f i c i e n t s

independent,

By d e f i n i t i o n ,

~k o f the expansion

system. Let us d e t e r m i n e f > 0 iff

n tk n-k f = ~ ~k (1 - t ) are nonk=0

negative.

( c ) We show t h a t f > O and g > 0 i m p l y fg > O. Indeed, f =

m ~ k=O

>'k t k (1 -

t)m- k

~k -> 0 , k = O . . . . . m,

n

g =

fg =

~ ~ ~=0 m+n ~ j =0

k+~ =j

t L ( I - t ) n-~

(

~

>_ O,

~= O . . . . . n,and

~ Xk ~g) t J ( 1 - t ) m+n-j k+9~=j

~k ug~ >- 0 . . . . , m+n.

with

for sufficiently

150 (d) Let f E Z [ t ]

such t h a t f ( t )

m

> 0 for t ~ [0,I], ~

then f > O. We w r i t e

t-!

f = Z apt p and fn = ~ apt p=O p=O

t - P--!

n i - 1 n

. . n n > m. ' i - p-1 ' n

Since fn converges to f uniformly on [0,1]~ f o r n s u f f i c i e n t l y l a r g e , f n ( t ) > 0 f o r ~-E [0,1] .- Th~n-,. . . . . . . . . . . . .

m f = ~

: ~ k=O m

x

p=O

apt p =

m ~

~ ap p=O

n ao

(I

k-

(1 - t)

n-p p=O

, and

k(k-1)...(k-p+l) = (~)fn n(n-1) (n-p+1)

(e) We conclude t h a t f o r a non zero f e ~ [ ~ ,

f > 0 iff

The c o n d i t i o n is c l e a r l y necessary. To show t h a t i t f = t m g(1 - t ) n with g ( t ) ~ 0 f o r t ~ [ 0 , I ] .

f(t)

(~) > O.

> 0 for t 00,1[.

is s u f f i c i e n t ,

By (d), g

write

> 0 and by ( c ) , f > O.

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NOTATION INDEX (References are to the pages on which the symbols are d e f i n e d ) . group of complex numbers. Q group of r a t i o n a l

EXAMPLES

numbers

IR group of real numbers

CAR

129

T group of complex numbers of modulus 1

GICAR

130

On

140

On

141

I group o f integers GROUPOID THEORY

HAAR SYSTEMS AND MEASURES x, y, z . . . .

elements o f the groupoid G

u, v, w. . . .

u n i t s o f the groupoid G

S,T . . . .

or s , t . . . .

d(x), r(x)

G-sets or G-maps

I0

xs, sx

10

u.s AB, A- I GO, G2

10 6 6

Gu, Gv, Gvu ~ G(u) 6

35

GE or GI E S1 × G2

8 122

G1 m G2 G(c)

122 8

Gx~A

73

H\G Gn

75 11

Cn(G,A), Zn(G,A),Bn(G,A), Hn(G,A) £G(A), £(a) Ext(A,G)

gc

[4

z4

~(. ,s)

z9

&(-,s)

29

one-cocycle

Min(c)

27

R(c), R~(C) RU(c),R~(C)

36 36

T(c)

37

Rl(C ) 45 T-valued two-cocycle 15 12

12 13

Ko(G ), D(G) 131 ~b'

24 23

c

8

Ga

VO D

COCYCLES

[u]

[A]

16 17

v, ~2, v-1 22

6

d(s), r(s)

{lm} {(~2) x }

33

gn 14 c n ( g , 4 ) , z n ( g , ~ ) , Bn(g,~), Hn(g,~) 15

3ROUPOID ALGEBRAS Cc(X ) B(G)

16 61

Cc(G,~ ) B(G,~)

48 61

f,g

f u n c t i o n s on the groupoid

h

f u n c t i o n on the u n i t space GO

G

156

GROUPOID ALGEBRAS ( c o n t i n u e d ) f.

g

48

f * sf,

48 fs,

s * f

62

hf

59

hs

64

rlflli,,~ , llelli,d ' I1flli 50 [Ifll 51 ]lfllred 82 C*(G,d),

C*red ~(B),

C*(G)

(G,~) g(B),

58 82

qj~(B)

104

sa

104

r(~)

112

f • ¢, ¢ • f , < f ' g > B' < f ' g > Ind,~, IndM

,@ • f , E

f

• ~

77 78 82

SUBJECT INDEX (The f i r s t reference is to the page of these notes where the expression is defined ; the following references are to a r t i c l e s where a similar notion appears ; they are intended only to serve as a guide to the subject ; standard references to C*-algebra theory are [ 1 ~ , [ 6 4 ] and [60]).

Almost invariant set 24, ~1] 274 amenable groupoid 92, ~1] 354 amenable quasi-invariant measure 86

ample semi-group 2O,

[2 119

ample semi-group of an abelian sub C~-algebra

i04,[ 2]

approximately elementary groupoid 123 approximately f i n i t e groupoid 123, [5~ asymptotic range 36, [31] 317, [49] Borel G-set 33 bounded representation C-bundle

51

11, ~9110

C-sheaf 14 C*-algebra of a groupoid 58, ~4] 35 Cartan subalgebra 106, 135, ~ i ] 335 coboundary 12 cocycle 12 cohomology group 12, ~6] 467 continuous G-set 33, 38 convolution product 48, ~5] 624 Cuntz algebra 145, ~5], ~0] Cuntz groupoid 140 Cuntz inverse semi-group. 141 Dimension group of an AF groupoid 131, dimension range of an AF groupoid 131 d i s j o i n t union of groupoids domain 6,10 Elementary groupoid 123 Elliott group 132, [27], [25]

122

[52] • rL27] 25

158 energy cocycle 116 energy operator 115 equivalence relation 7, 17, 22, 34, ~1] ergodic measure 24, ~ I ] 274 e s s e n t i a l l y principal groupoid 100 extension 12, ~6] 128 extension groupoid 73, ~4] 105 Finite idempotent element f i n i t e type groupoid 131

131

G-bundle I I , [79110 G-map I0 G-module bundle 11, [79] 10 G-set i0 g-sheaf 14 gauge automorphism group 129, [8] 227 Glimm groupoid 128, [35] ground state 27, [65] 98 group bundle 7, [79] 8 groupoid 5, [44]3, [53], [611256, [79] Haar system 16, [68] 27, [77]2 homomorphism 7, [44] 4 horizontal Radon-Nikodym derivative

29

Induced measure 22, [31] 293 induced representation 81, [63] inductive l i m i t of groupoids 122 i n f i n i t e type groupoid 131 invariant mean (of a measure groupoid) 91, [83] 30 ~nvariant measure 27, [31] 293 Inverse semi-group 20, [52] 2 involution 48,[75] 625 irreducible groupoid 35 Ising model 126, [33] isotropy group 6 KMS condition 27, [73] 63

159 Measurewise amenable groupoid 92, [81] 354 minimal groupoid 35, [24]7 modular function 24, [44114, [311293 Non-singular G-set 33, [51] normalizer of an abelian subalgebra 104, ~11332, [17] Orbit

6

Partial isomorphism 14 physical ground state 27, [651100 principal groupoid 6 , ~ i ] product of groupoids 122 product type cocycle 128, [9] properly ergodi¢ measure 26, [61] 278 Quasi-invariant measure 23, [31] 291 quasi-orbit 26, [5~ 447 r-discrete groupoid 18,[31] Radon-Nikodym derivative 24, [31] 293 range 6, I0 range of a cocycle 36 reduced C*-algebra of a groupoid 82 reduction of a groupoid 8, [44] 3 regular abelian subalgebra 104, [31] 332, [17] regular representation 55, [45] 54 r e l a t i v e l y free action 21 representation of an inverse semi~group 143, [1] 363 representation of Cc(G,~) 50, [75] 626 representation of G 52, [7~ 626, [45]47 o-representation 52 a-regular representation 55, [45] 54 saturation of a measure 25 saturation of a set 35 semi-direct product 8, 96 s i m i l a r i t y 7, [6~ 259

skew-product

8, 93, [31] 315, [53]

s u f f i c i e n t l y many non-singular Borel G-sets 33

160 T-set of a cocycle 37, [13] 152 topological groupoid 16, [26] 137 transformation group 6, 17, 22, 34, [24] t r a n s i t i v e groupoid 6, [75] t r a n s i t i v e measure 26, [61] 277 type I groupoid 27, [36] type I, II 1, II , I l l quasi-orbit 27, [55] 447 Unit 6 unit space

6

Vertical Radon-Ni~odym derivative 29