Invariant states and asymptotic abelianness

Abstract

Given a C ∗-algebra acted upon by a group of automorphisms, we study quite general conditions which allow to deduce the main properties of asymptotically abelian systems, of interest in Quantum Statistical Mechanics. For von Neumann algebras, our conditions imply in particular the existence of a unique normal invariant projection onto the fixed points of the center. We also describe the interrelations between the main notions of asymptotic abelianness so far introduced.