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.