GROUPOID C""ALGEBRAS 1 Introduction 2 Definitions and notation

96 M¼ad¼alina Roxana BuneciTheorem 14. Let G be a locally compact second countable groupoid with properorbit space. Let u ; u 2 G (0) be a Haar systems on G. Let F i , i = 1; 2, be twoBorel subsets of G (0) containing only one element e i (u) in each orbit [u]. For eachi = 1; 2, let i : G (0) ! G F ibe a cross section for d Fi : G F i! G (0) , d Fi (x) = d (x),satisfying the conditions1. i (e i (v)) = e i (v) for all v 2 G (0)2. i (K) is relatively compact in G for all compact sets K G (0) .Then the C -algebras M 1(G; ) and M 2(G; ) are -isomorphic. (Theorem 6[8]).Thus M (G; ) is a C -algebra which does not depend on the choice of the Haarsystem and also does not depend on the choice of cross section .Theorem 15. Let G be a locally compact second countable locally transitive groupoidendowed with a Haar system u ; u 2 G (0) . Let F be a subset of G (0) containingonly one element e (u) in each orbit [u]. Let : G (0) ! G F be a regular cross sectionof d F . ThenC (G; ) = M (G; ) = M (G; ) .(Proposition 18 [7])Theorem 16. Let G be a locally compact second countable principal proper groupoid.Let F be a Borel subset of G (0) meeting each orbit exactly once. Let : G (0) ! G Fbe a cross section for d : G F ! G such that (K) is relatively compact in G for allcompact K G (0) . Let u ; u 2 G (0) be a Haar system on G. Then(Corollary 23 [7]).ReferencesC (G; ) M (G; ) M (G; ) .[1] C. Anantharaman-Delaroche, J. Renault, Amenable groupoids, Monographiede L’Enseignement Mathematique No 36, Geneve, 2000.MR1799683(2001m:22005). Zbl 0960.43003.[2] W. Arveson, An invitation to C -algebras, Springer-Verlag, New York, 1976.MR0512360(58 #23621). Zbl 0344.46123.[3] L.G. Brown, P. Green, M. Rie¤el, Stable isomorphism and strong Moritaequivalence of C -algebras, Paci…c J. Math. 71(1977), 349-363. MR0463928(57#3866). Zbl 0362.46043.******************************************************************************Surveys in Mathematics and its Applications 1 (2006), 71 –98http://www.utgjiu.ro/math/sma