Amenable Groups and Generalized Cuntz Algebras

WithHa Hilbert space, OHthe generalized Cuntz algebra overHendowed with the canonical actionΛofU(H), we consider a natural family S0of product representations of the zero grade part O0H, and the family S obtained from S0by inducing to OHvia the natural conditional expectationP: OH→O0H. Each element of S0(and hence of S) is labelled by a sequence of unit vectors inH. LetGbe a non-compact locally compact group,λits regular representation onH=L2(G), and OHendowed with the actionΛ∘λofG; we prove thatGisσ-amenable iff there are ireducible covariant representations in S.