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

By Jean Renault

ISBN-10: 3540099778

ISBN-13: 9783540099772

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The f o l l o w i n g p r o p o s i t i o n gives some basic p r o p e r t i e s of the q u a n t i t i e s R (c) and T(c) ; in p a r t i c u l a r , they depend o n l y on the cohomology class o f c. The aim o f t h i s section is to show t h e i r usefulness, j u s t i f y i n g t h e i r i n t r o d u c t i o n . Further references to the asymptotic range and the T-set o f a cocycle can be found in [31] in the context of ergodic theory. 5. Proposition : introof tensor products of f a c t o r s o f type I.

8 ( i i ) . Suppose t h a t there e x i s t s an open neighborhood V o f 0 in A, u e G0, a base of neighborhoods of u and a net {x U} such t h a t x u ~ GlU and c(x u) ~ V. 41 I f {a U} is a subset of {C(Xu)} converging to a, then a # V and a E n R(Cu) where the i n t e r s e c t i o n is taken over a base of neighborhoods of u. 9, a c R (c). Since R (c) = { 0 } , t h i s is a c o n t r a d i c t i o n . D. Note t h a t i f A is t o r s i o n f r e e , the c o n d i t i o n R(c) compact already implies R (c) = {0~.

Let u be a measure on the u n i t space o f a second g r o u p o i d G. Let E be a Borel set in G o f p o s i t i v e v-measure. Since G can be covered by c o u n t a b l y many open G - s e t s , t h e r e e x i s t s an open G-set t such t h a t s = E of positive (c) t has p o s i t i v e ~-measure. Then, s i s a n o n - s i n g u l a r Borel G-set u-measure. Transitive the t r a n s i t i v e principal g r o u p o i d s . Let × be a l o c a l l y compact space. We d e f i n e g r o u p o i d on the space X as G = X x X, w i t h the g r o u p o i d s t r u c t u r e g i v e n i n 1 .

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A groupoid approach to C* - algebras by Jean Renault


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