A groupoid approach to C* - algebras by Jean Renault

By Jean Renault

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Proof : Suppose t h a t a c R(Cu) f o r every U in a base of neighborhoods of uO. Let V,W be neighborhood o f e on A such t h a t W + W c V and U be a non-empty open set. There e x i s t s x ~ G w i t h r ( x ) = u O, d(x) c U and a continuous G-set s c o n t a i n i n g x. ~• Let z = s - l y s , c(z) = c ( s - l r ( y ) ) + c(y) + c(d(y)s) ~ a + W+ then z ~ Gld(s ) c GIU and Wc a + V• Thus, a c R(Cu) f o r any non-empty open set U. Q•E •D. The f o l l o w i n g theorem may be compared w i t h theorem 9 o f [ 3 1 , 1 ] .

Proof : Let a(x) = d(vs-1) (x) be the v e r t i c a l Radon-Nikodym d e r i v a t i v e of s with - d~ respect to ~. Since vs " I = f d ( s ) (~Us-1) dr(u) and vs - I = f d ( s ) ~ ( ~ u ) d~(u) are two r-decompositions of vs -1, there e x i s t s a ~-conull set U in GO such that f o r every u in U, ~u s-1 = ~xu. e. x in d - l [ r ( s ) ] , ~(x) = d(~Us-1)(x). d~u The commutativity of l e f t and r i g h t m u l t i p l i c a t i o n allows us to w r i t e , f o r any x in GU and any p o s i t i v e measurable f , f f ( y ) ~ ( x y ) d~d(X)(y) = f f ( x - l y ) { ( y ) d~r(X)(y) = ~ f ( x - l y s -1) d~r(X)(y) = ~f(ys -1) dxd(X)(y) = ff(y) ~(Y) dxd(X)(y) .

The set {v c GO : f o r any b e A, ( v , b ) e F} is non-empty, G - i n v a r i a n t and closed. Since G is m i n i m a l , t h i s i s GO , hence F = GO × A. D. 15. 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 w i t h open range map, A a t o p o l o - g i c a l group and c ~ Z I ( G , A ) . 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 .

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