States and automorphism groups of operator algebras

Suppose that a group of automorphisms of a von Neumann algebraM, fixes the center elementwise. We show that if this group commutes with the modular (KMS) automorphism group associated with a normal faithful state onM, then this state is left invariant by the group of automorphisms. As a result we obtain a “noncommutative” ergodic theorem. The discrete spectrum of an abelian unitary group acting as automorphisms ofM is completely characterized by elements inM. We discuss the KMS condition on the CAR algebra with respect to quasi-free automorphisms and gauge invariant generalized free states. We also obtain a necessary and sufficient condition for the CAR algebra and a quasi-free automorphism group to be η-abelian.