GROUPOID C""ALGEBRAS 1 Introduction 2 Definitions and notation

Groupoid C -algebras 73A groupoid is said transitive if and only if it has a single orbit, or equivalently ifthe map : G ! G (0) G (0) , de…ned by (x) = (r (x) ; d (x)) for all x 2 G,is surjective. Generally, for each orbit [u] of a groupoid G, the reduction of G to[u], Gj [u] , is a transitive groupoid called transitivity component of G. It is easy tosee that (algebraically) G is the disjoint union of its transitivity components. Agroupoid is said principal if the map (de…ned above) is injective.By a homomorphism of groupoids we mean a map ' : G ! (with G,groupoids) satisfying the following condition:if (x; y) 2 G (2) , then ('(x); '(y)) 2It follows that '(x 1 ) = ('(x)) 1 and '(G (0) ) (0) .2.2 Borel groupoids and topological groupoids(2) and '(xy) = '(x)'(y)We shall state some conventions and facts about measure theory (see [2], Chapter3).By a Borel space (X; B (X)) we mean a space X, together with a -algebraB (X) of subsets of X, called Borel sets. A subspace of a Borel space (X; B (X)) isa subset S X endowed with the relative Borel structure, namely the -algebra ofall subsets of S of the form S \ E, where E is a Borel subset of X. (X; B (X)) iscalled countably separated if there is a sequence (E n ) nof sets in B (X) separatingthe points of X: i.e., for every pair of distinct points of X there is n 2 N such thatE n contains one point but not both.A function from one Borel space into another is called Borel if the inverse imageof every Borel set is Borel. A one-one onto function Borel in both directions is calledBorel isomorphism.The Borel sets of a topological space are taken to be the -algebra generatedby the open sets. (X; B (X)) is called standard if it is Borel isomorphic to a Borelsubset of a complete separable metric space. (X; B (X)) is called analytic if it iscountably separated and if it is the image of a Borel function from a standard space.The locally compact Hausdor¤ second countable spaces are analytic.By a measure on a Borel space (X; B (X)) we always mean a map : B (X) !R which satis…es the following conditions:1. is positive ( (A) 0 for all A 2 B (X))2. (;) = 0 1SP3. is countable additive (i.e. A n = 1 (A n ) for all sequences fA n g nn=1 n=1of mutually disjoint sets A n 2 B (X))******************************************************************************Surveys in Mathematics and its Applications 1 (2006), 71 –98http://www.utgjiu.ro/math/sma