With H a Hilbert space, O H the generalized Cuntz algebra over H endowed with the canonical action 4 of U(H), we consider a natural family S 0 of product representations of the zero grade part O 0 H , and the family S obtained from S 0 by inducing to O H via the natural conditional expectation P: O H ร O 0 H . Each element of S 0 (and hence of S) is labelled by a sequence of unit vectors in H. Let G be a non-compact locally compact group, * its regular representation on H=L 2 (G), and O H endowed with the action 4 b * of G; we prove that G is _-amenable iff there are ireducible covariant representations in S.
1997 Academic Press 0. INTRODUCTION Throughout this paper, G will be a topological locally compact group. For some notations, look at the end of this introduction. By definition, G is called amenable iff [Pi, G] there exists a left invariant state on the C*-algebra CB(G) of all continuous bounded C-valued functions on G or, equivalently, on the W*-algebra L (G) of all (equivalence classes of) complex Haar measurable essentially bounded functions on the group. It is well know that abelian, compact, or solvable locally compact groups are amenable. On the contrary, every non-compact semisimple Lie group is not amenable.