GROUPOID C""ALGEBRAS 1 Introduction 2 Definitions and notation

Groupoid C -algebras 77with X. Two extreme cases deserve to be single out. If E = X X, then Eis called the trivial groupoid on X, while if E = diag (X), then E is called theco-trivial groupoid on X (and may be identi…ed with the groupoid in example3).If G is any groupoid, thenR = f(r (x) ; d (x)) ; x 2 Ggis an equivalence relation on G (0) . The groupoid de…ned by this equivalencerelation is called the principal groupoid associated with G.Any locally compact principal groupoid can be viewed as an equivalence relationon a locally compact space X having its graph E X X endowedwith a locally compact topology compatible with the groupoid structure. Thistopology can be …ner than the product topology induced from X X. We shallendow the principal groupoid associated with a groupoid G with the quotienttopology induced from G by the map : G ! R; (x) = (r (x) ; d (x))This topology consists of the sets whose inverse images by in G are open.2.4 Haar systemsFor developing an algebraic theory of functions on a locally compact groupoid, oneneeds an analogue of Haar measure on locally compact groups. Several generalizationsof the Haar measure to the setting of groupoids were taken into considerationsin the literature (see [27], [23], [10], [9], [19]). We use the de…nition adopted byRenault in [19]. The analogue of Haar measure in the setting of groupoids is a systemof measures, called Haar system, subject to suitable invariance and smoothnessconditions called respectively "left invariance" and "continuity". More precisely, a(left) Haar system on a locally compact Hausdor¤ groupoid G is a family of positiveRadon measures (or equivalently, Borel regular measures) on G, = u ; u 2 G (0) ,such that1) For all u 2 G (0) , supp( u ) = G u .2) For all f 2 C c (G),Zu 7!f (x) d u (x)h i: G (0) ! Cis continuous.3) For all f 2 C c (G) and all x 2 G,Zf (y) d r(x) (y) =Zf (xy) d d(x) (y)******************************************************************************Surveys in Mathematics and its Applications 1 (2006), 71 –98http://www.utgjiu.ro/math/sma