GROUPOID C""ALGEBRAS 1 Introduction 2 Definitions and notation

Groupoid C -algebras 91It can be shown that kxk = khx; xik 1 2 de…nes a norm on X. If X is completewith respect to this norm, it is called a Hilbert A-module. If not, all the structurecan be extended to its completion to turn it into a Hilbert A-module.As a normed linear space, a right Hilbert A-module X carries an algebra ofbounded linear transformations. In the following we shall denote by B (X) thecollection of all bounded linear maps T : X ! X which are module maps (this meansthat T (x a) = T (x)a for all x 2 X, a 2 A) and adjointable (this means that thereis another linear bounded operator T on X such that hT (x) ; yi A= hx; T (y)i Aforall x; y 2 X). It is easy to see that if X is Hilbert space and A = C (the space ofcomplex numbers), then B (X) is the algebra of linear bounded operators on X. Thealgebra of linear bounded operators on a Hilbert X has a two-sided closed non-trivialideal K (X) (the compact operators). The analog of that ideal of compact operatorsis given in the case of a right Hilbert A-module X by the closed linear span in B (X)of all the "rank one" transformations on X, i.e. of all transformations of the formx y : X ! X de…ned by x y (z) = x hy; zi Afor all z 2 X, with x; y 2 X. Theclosed linear span in B (X) of all transformations xy (with x; y 2 X) is denoted byK (X), it is called the imprimitivity algebra of X and its elements are called compactoperators on X. If X is a left Hilbert A-module, then imprimitivity algebra K (X)is the closed linear span of all transformations of the form x y : X ! X de…nedby x y (z) = A hz; yi x for all z 2 X, with x; y 2 X.Proposition 6. If A is a C -algebra and if X is a right Hilbert A-module, then B (X)and K (X) are C -algebras, with B (X) equal to the multiplier algebra of K (X).Further, X becomes a left Hilbert C -module over K (X), where the K (X)-valuedinner product is de…ned by the formula:K(X) hx; yi = x y , for all x; y 2 X,and K (X) (X as a left Hilbert C -module over K (X)) is naturally isomorphic to Athough the formulax y ! hx; yi A.De…nition 7. Let A and B be C -algebras. By an (A; B)-equivalence bimodule wemean an A, B -bimodule X equipped with A and B-valued inner products with respectto which X is a right Hilbert B-module and a left Hilbert A-module such that:1. A hx; yi z = x hy; zi Bfor all x; y; z 2 X.2. hax; axi B kak 2 hx; xi Bfor all a 2 A, x 2 X and A hxb; xbi kbk 2 Ahx; xi forall b 2 B, x 2 X.3. hX; Xi Bspans a dense subset of B and A hX; Xi spans a dense subset of A.******************************************************************************Surveys in Mathematics and its Applications 1 (2006), 71 –98http://www.utgjiu.ro/math/sma