By Jean Renault

ISBN-10: 3540099778

ISBN-13: 9783540099772

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**Extra resources for A groupoid approach to C* - algebras**

**Sample text**

A homeomorphism onto an open subset of GO]. (ii) Suppose t h a t G has a Haar system {xu}. A n o n - s i n g u l a r Borel G-set [resp. non-singular continuous G-set] is a Borel G-set [resp. a continuous G-set] such t h a t there e x i s t s a Borel [resp. e. x c d - 1 [ r ( s ) ] . e. x c d - l [ r ( s ) ] . 25. Examples :_ In the case of a t r a n s f o r m a t i o n group (U,S), the G-set s = {(u,u • s) : u ~ V } where V is an open subset of U and s E S, is a n o n - s i n g u l a r continuous G-set.

I) A G-set s w i l l restriction be c a l l e d a Borel G-set [resp. a continuous G-set I i f the of each of the maps r and d to s is a Borel isomorphism onto a Borel subset of GO [resp. a homeomorphism onto an open subset of GO]. (ii) Suppose t h a t G has a Haar system {xu}. A n o n - s i n g u l a r Borel G-set [resp. non-singular continuous G-set] is a Borel G-set [resp. a continuous G-set] such t h a t there e x i s t s a Borel [resp. e. x c d - 1 [ r ( s ) ] . e. x c d - l [ r ( s ) ] . 25.

In the case of a r - d i s c r e t e groupoid, any open G-set s is a n o n - s i n g u l a r continuous G-set. We have already observed t h a t i t s v e r t i c a l Radon-Nikodym d e r i v a t i v e 5(u,s) is equal to 1, f o r u c r ( s ) . 26. The set o f n o n - s i n g u l a r Borel G-sets [ r e s p . n o n - s i n g u l a r continuous G-sets] is an inverse semi-group under the operations ( s , t ) ÷ st and s ÷ s -1. We c a l l i t the Borel ample semi-group o f G and denote i t and w r i t e ~c ] .

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

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