Asymptotically abelian systems

Abstract

We study pairs {\(\mathfrak{A}\), α} for which\(\mathfrak{A}\) is aC*-algebra and α is a homomorphism of a locally compact, non-compact groupG into the group of *-automorphisms of\(\mathfrak{A}\). We examine, especially, those systems {\(\mathfrak{A}\), α} which are (weakly) asymptotically abelian with respect to their invariant states (i.e. 〈Φ |A α g (B) — α g (B)A〉 → 0 asg → ∞ for those states Φ such that Φ(α g (A)) = Φ(A) for allg inG andA in\(\mathfrak{A}\)). For concrete systems (those with\(\mathfrak{A}\)-acting on a Hilbert space andg → α g implemented by a unitary representationg →U g on this space) we prove, among other results, that the operators commuting with\(\mathfrak{A}\) and {U g } form a commuting family when there is a vector cyclic under\(\mathfrak{A}\) and invariant under {U g }. We characterize the extremal invariant states, in this case, in terms of “weak clustering” properties and also in terms of “factor” and “irreducibility” properties of {\(\mathfrak{A}\),U g }. Specializing to amenable groups, we describe “operator means” arising from invariant group means; and we study systems which are “asymptotically abelian in mean”. Our interest in these structures resides in their appearance in the “infinite system” approach to quantum statistical mechanics.