Download e-book for kindle: A groupoid approach to C* - algebras by Jean Renault

By Jean Renault

ISBN-10: 3540099778

ISBN-13: 9783540099772

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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.

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.

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.

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A groupoid approach to C* - algebras by Jean Renault


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