GROUPOID C""ALGEBRAS 1 Introduction 2 Definitions and notation

Groupoid C -algebras 891. ( x) = r () for all (; x) 2 X.2. (x) x = x for all x 2 X.3. If ( 2 ; 1 ) 2 (2) and ( 1 ; x) 2 X, then ( 2 1 ) x = 2 ( 1 x).If is a topological groupoid and X is a topological space, then we say that a leftaction is continuous if the map is continuous and open and the map (; x) 7! xis continuous, where X is endowed with the relative product topology comingfrom X.The action is called free if (; x) 2 X and x = x implies 2 (0) .The continuous action is called proper if the map (; x) 7! ( x; x) from Xto X X is proper (i.e. the inverse image of each compact subset of X X is acompact subset of X).In the same manner, we de…ne a right action of on X, using a continuous map : X !(0) and a map (x; ) 7! x fromX = f(x; ) : (x) = r () gto X.The simplest example of proper and free action is the case when the locallycompact Hausdor¤ groupoid acts upon itself by either right or left translation(multiplication).De…nition 3. Let 1; 2 be two groupoids and X be set. Let us assume that 1 actsto the left on X with momentum map : X ! (0)1 , and that 2 acts to the right onX with momentum map : X ! (0)2 . We say that the actions commute if1. (x 2 ) = (x) for all (x; 2 ) 2 X 2 and ( 1 x) = (x) for all ( 1 ; x) 21 X:2. 1 (x 2 ) = ( 1 x) 2 for all ( 1 ; x) 2 1 X, (x; 2 ) 2 X 2 .De…nition 4. Let 1; 2 be two locally compact Hausdor¤ groupoids having openrange maps. The locally compact Hausdor¤ space X is said a ( 1 ; 2)-Moritaequivalence if the following conditions are satis…ed:1. 1 acts to the left on X with momentum map : X ! (0)1 and the action iscontinuous free and proper.2. 2 acts to the right on X with momentum map : X ! (0)2 and and theaction is continuous free and proper3. The actions commute.******************************************************************************Surveys in Mathematics and its Applications 1 (2006), 71 –98http://www.utgjiu.ro/math/sma