Asymptotic invariance, amenability and generalized Cuntz algebras

Abstract

Given a unitary representation u of the locally compact group G on the Hilbert space H, we investigate the notion of infinite tensor power of u. Then we apply the results to study covariance of the associated canonical action of G on the generalized Cuntz algebra $\mathcal{O}_H$ in the GNS representation of (the gauge-invariant extension of) some pure quasi-free state. We examine in detail the case of a measure-preserving action of G on a measure space X. In this case, covariance is (almost) equivalent to the existence of a G-invariant state on $L^\infty(X)$.