By Jean-Paul Pier

ISBN-10: 0471893900

ISBN-13: 9780471893905

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**Additional resources for Amenable locally compact groups **

**Sample text**

34 . X $ E Ax fl B' cause $ = E X , and $ = ~ ~ ( $ 1But . i n t h i s case $ E E ( A x n B ' ) t ~ ~ ( $ ) E EiB;) and c X is a n i s o m e t r y . 35 PROOF J, E E ( B ' ) . A s s u m e , by c o n t r a d i c t i o n , t h a t $ ft E ( B ' ) . , Let f < 1 be given. Then E E E L with U = {t E X ; I If(t) 1 1 < € 1 is open and contains x. Let g E C ( X ) with 0 5 g 5 1, g(x) = 1 and g(t) = 0 for t P: U be chosen Since I / @ 1/ = 1, there is v E Ex such that 1 IvI I < 1, @ ( v ) is real Let and @(v) > 1 - E.

By Namely t h a t Theorem 1 . 2 6 above i m p l i e s B i s h q ' s Theorem. To do s o w e w i l l u s e B i s h o p ' s o r i q i n a l d e s c r i p t i o n t h e p a r t i t i o n o f X i n t o a n t i s y m m e t r i c sets ( S e e [8] 1. I f X i s a H a u s d o r f f s p a c e and A C C ( X ; I K ) salqebra, is b e t h e class of a l l o r d i n a l numbers let c a r d i n a l numbers a r e less o r e q u a l t o 2 "I, a whose is where ' X I of c a r d i n a l number o f X . F o r e a c h Q E , w e d e f i n e by f i n i t e induction a closed, pair-wise d i s j o i n t coverina the transof X, d e n o t e d by P a .

C ( X ; I R 1 b e t h e s u b a l g e b r a c o n s i s t i n g of a l l f u n c t i o n s of the form ( t , Z ) E X -+ 9 q ( t ) , where q E F f ( E ; I R ) . Obviously, is an A-module. For each t E E , such t h a t K t # 0 , l e t Xt = { ( t , Z ) E X ; Z E Kt). s e p a r a t e s t h e p o i n t s o f E , t h e e q u i v a l e n c e classes Y Since c X E' (mod. are p r e c i s e l y t h e sets X t d e f i n e d above. I n t h e n e x t Theorem w e s h a l l suppose t h a t X i s comp a c t and f o r e a c h t E E,Kt i s a c a n p a c t s u b s e t o f CC which h a s a c o n n e c t e d complement.

### Amenable locally compact groups by Jean-Paul Pier

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