A groupoid approach to C* - algebras by Jean Renault

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By Jean Renault

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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 ) .

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 ) .

D. 11, R (c) = R1(c ) = T(c) i . 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. 2) chosen once f o r a l l . 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. g. 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.

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