By Jean Renault
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X (eod(x)) - I with o : r[#O(x)] die(u)] = u #0 (u) c O. D. Let G be a t o p o l o g i c a l l i a n . 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.
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. 3. 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.
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 ) .
A groupoid approach to C* - algebras by Jean Renault