A groupoid approach to C* - algebras by Jean Renault PDF

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By Jean Renault

ISBN-10: 3540099778

ISBN-13: 9783540099772

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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 ) .

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.

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 . 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.

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A groupoid approach to C* - algebras by Jean Renault

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