Are interacting systems norm asymptotic abelian?

Abstract

We consider infinite Fermi systems for which time evolution is defined as automorphism group. Under appropriate assumptions quasifree evolutions are norm asymptotic abelian on the even subalgebra. If interaction is added in the ground state strong asymptotic abelianess is preserved. In the tracial state, however, it is argued that for appropriate interacting systems even strong asymptotic abelianess fails in first order perturbation theory and only weak asymptotic abelianess seems to be possible. Consequences of different kinds of abelianess are discussed.