Asymptotically abelian systems

Abstract

We study pairs { {Mathematical expression}, α} for which {Mathematical expression} is a C*-algebra and α is a homomorphism of a locally compact, non-compact group G into the group of *-automorphisms of {Mathematical expression}. We examine, especially, those systems { {Mathematical expression}, α} which are (weakly) asymptotically abelian with respect to their invariant states (i.e. 〈Φ |A αg(B) - αg(B)A〉 → 0 as g → ∞ for those states Φ such that Φ(αg(A)) = Φ(A) for all g in G and A in {Mathematical expression}). For concrete systems (those with {Mathematical expression}-acting on a Hilbert space and g → αg implemented by a unitary representation g →Ug on this space) we prove, among other results, that the operators commuting with {Mathematical expression} and {Ug} form a commuting family when there is a vector cyclic under {Mathematical expression} and invariant under {Ug}. 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 { {Mathematical expression}, Ug}. 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. Show more