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

By Jean Renault

ISBN-10: 3540099778

ISBN-13: 9783540099772

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In undemanding introductions to mathematical research, the therapy of the logical and algebraic foundations of the topic is unavoidably really skeletal. This ebook makes an attempt to flesh out the bones of such therapy by means of offering a casual yet systematic account of the rules of mathematical research written at an common point.

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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 .