GROUPOID C""ALGEBRAS 1 Introduction 2 Definitions and notation

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92 M¼ad¼alina Roxana BuneciWe call A and B strongly Morita equivalent if there is an (A; B)-equivalencebimodule.Strong Morita equivalence is an equivalence relation.If A is a C -algebra and if X is a Hilbert A-module, then A and K (X) arestrongly Morita equivalence.Proposition 8. Let A and B be C -algebras and X be an (A; B)-equivalence bimodule.Then the map from K (X) (X as a right Hilbert C -module over B) to Ade…ned by formulaAx y 7! hx; yiis a C -isomorphism. Similarly, the map from K (X) (X as a left Hilbert C -moduleover A) to A de…ned by formulais a C -isomorphism.x y 7! hx; yi BThus, two C -algebras are strongly Morita equivalent if and only if one can berealized as compact operators of a Hilbert C -module over the other.Theorem 9. If A and B are two stably isomorphic C -algebras (in the sense thatA K is isomorphic to B K, where K is the algebra of compact operators on aseparable Hilbert space), then A and B are strongly Morita equivalent. Conversely, ifA and B have countable approximate identities, and if A and B are strongly Moritaequivalent, then A and B are stably isomorphic. ([3])Using the correspondence of groupoid representations and representations of C -algebra, Muhly, Renault and Williams proved the following theorem:Theorem 10. Let and G be locally compact, second countable, Hausdor¤ groupoidsendowed with the Haar systems and , respectively. If and G are Morita equivalent,then C ( ; ) and C (G; ) are strongly Morita equivalent. (Theorem 2.8/p.10 [13]).5 Results concerning the independence of the groupoidC -algebra of the Haar systemThe de…nition of the groupoid C -algebra depends on the choice of the Haar system(the convolution is de…ned using a Haar system). In the group case, Haar measure isessentially unique, but for groupoids, this is no longer the case. Due to Theorem 10,di¤erent choices of Haar system produces strongly Morita equivalent C -algebras.This still leaves open the question: are the C -algebras associated with two Haarsystems -isomorphic. As Muhly, Renault and Williams proved, this is indeed thecase for transitive groupoid:******************************************************************************Surveys in Mathematics and its Applications 1 (2006), 71 –98http://www.utgjiu.ro/math/sma
Groupoid C -algebras 93Theorem 11. If the locally compact, second countable, Hausdor¤ groupoid G istransitive, then the (full) C -algebra of G is isomorphic to C (H) K L 2 () ,where H is the isotropy group G e e at any unit e 2 G (0) , is an essentially uniquemeasure on G (0) , C (H) denotes the group C -algebra of H, and K L 2 () denotesthe compact operators on L 2 (). (Theorem 3.1/p.16 [13, Theorem 3.1, p. 16]).Consequently, the (full) C -algebras of a transitive locally compact, secondcountable, Hausdor¤ groupoid G associated with two Haar systems are -isomorphic.In order to prove that result Muhly, Renault and Williams …rstly established thatC (G) and C (H) are strong Morita equivalent via a (C (H) ; C (G)) -equivalencebimodule module X 1 (because G and H are Morita equivalent groupoids). As aconsequence the C -algebra of G is the imprimitivity algebra of X 1 . Then theyneeded another C (H) module X 2 (isomorphic to X 1 ) whose imprimitivity algebrais C (H) K L 2 () for a suitable measure .We can obtain the isomorphism between the C -algebra of G and C (H) K L 2 () more directly. If we endow G (0) H G (0) with the product topology,and the operations(u; x; v) (v; y; w) = (u; xy; w)(u; x; v) 1 = (v; x; u)then it becomes a locally compact second countable groupoid. The system of measures " u e ; u 2 G (0) , where " u is the unit point mass at u, e is the Haarmeasure on H = G e e and is a measure of full support on G (0) , is Haar system onG (0) HG (0) . It is not hard to prove that the C -algebra of G (0) HG (0) endowedwith the Haar system " u e ; u 2 G (0) is -isomorphic to C (H)K L 2 () .On the other hand since G is a locally compact, second countable, Hausdor¤ transitivegroupoid, the restriction of the domain map to G e is an open map (see [13]).According to Mackey Lemma (Lemma 1.1 [12]), there is : G (0) ! G e a regularBorel cross section of d : G e ! G (0) . This means that d ( (u)) = u for allu 2 G (0) and (K) has compact closure in G e for each compact set K in G (0) . Then : G ! G (0) H G (0) de…ned by (x) = r (x) ; (r (x)) x (d (x)) 1 ; d (x)is a Borel isomorphism which carries the Haar system of G into a Haar system ofG (0) H G (0) of the form " u e ; u 2 G (0) , where " u is the unit point massat u 2 G (0) , e is a Haar measure on H = G e e, and is a suitable Radon measureon G (0) with full support:Proposition 12. Let G be a locally compact second countable transitive groupoid.Let e be a unit and : G (0) ! G e be a regular Borel cross section of d : G e ! G (0) .Then : G ! G (0) G e e G (0) de…ned by (x) = r (x) ; (r (x)) x (d (x)) 1 ; d (x)******************************************************************************Surveys in Mathematics and its Applications 1 (2006), 71 –98http://www.utgjiu.ro/math/sma
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GROUPOID C""ALGEBRAS 1 Introduction 2 Definitions and notation

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