GROUPOID C""ALGEBRAS 1 Introduction 2 Definitions and notation

86 M¼ad¼alina Roxana Buneci2.3). The convolution of f, g 2 C c (X ) is given by:Zf g (x; ) = f (x; ) y; 0 g y; 0 1d (" x ) y; 0Z= f x; 0 g x 0 ; 0 1 d 0Z= f x; 0 g x 0 ; 0 1 d 0and the involution byf (x; ) = f (x; 1 ).3. If X is a locally compact Hausdor¤ space and if is a positive Radon measureon X with full support (i.e.supp () = X), then f" x ; x 2 Xg is a Haarsystem on X X (as a trivial groupoid). The convolution of f, g 2 C c (X X)is given by:f g (x; y) ==ZZf ((x; y) (t; z)) g (t; z) 1 d (" y ) (t; z)f (x; z) g (z; y) d (z)and the involution byf (x; y) = f (y; x).4. If X is a locally compact Hausdor¤ space, then f" x ; x 2 Xg is a Haar systemon X (seen as a co-trivial groupoid identi…ed with the groupoid described inExamples 3 Subsection 2.3). The convolution of f, g 2 C c (G) is given by:Zf g (x) = f (xy) g y 1 d" x (y)= f (xx) g x 1= f (x) g (x)and the involution byf (x) = f (x).Let = u ; u 2 G (0) be a Haar system on the locally compact Hausdor¤groupoid G. A representation of C c (G; ) is a -homomorphism L from the topological-algebra C c (G,) into B (H), for some Hilbert space H, that is continuouswith respect to the inductive limit topology on C c (G) and the weak operator topologyon B (H). The representation L is said non-degenerate if the linear span offL (g) : g 2 C c ( ) ; 2 Hg******************************************************************************Surveys in Mathematics and its Applications 1 (2006), 71 –98http://www.utgjiu.ro/math/sma