By Jean Renault

ISBN-10: 3540099778

ISBN-13: 9783540099772

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Let G be a t o p o l o g i c a l groupoid with open range map, A a t o p o l o - g i c a l group and c ~ ZI(G,A). Then (i) R (c) is a closed subgroup of A, T(c) is a subgroup of A, and R (c) and T(c) are orthogonal to each o t h e r . (ii) (iii) R (c) and T(c) depend o n l y on the class o f c. R (e) = {e} and T(e) = A, where e denotes the i d e n t i t y element of A as well as the constant cocycle e(x) = e. Proof : (i) Let us f i r s t show t h a t R(c) R (c) c R(c). Suppose a c R ( c ) and b e R (c).

The T-set of c is T(c) = {x E A : xoC E BI(G,~)}. 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.

One of the main j u s t i f i c a t i o n s f o r i t s i n t r o d u c t i o n , besides the need to deal with projective representations, is given in Section 4, where the C * - a l g e b r a of an r - d i s c r e t e p r i n c i p a l groupoid is characterized, under s u i t a b l e conditions, by the existence of a p a r t i c u l a r l y nice kind of maximal abelian subalgebra. One of these conditions is amenability, which is defined in Section 3. 48 An essential tool in the study of the C * - a l g e b r a of a groupoid is the correspondence, very f a m i l i a r in the case of a group, between the u n i t a r y representations of the groupoid and the non-degenerate representations of the C * - a l g e b r a .

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

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