States and automorphism groups of operator algebras

Suppose that a group of automorphisms of a von Neumann algebra M, fixes the center elementwise. We show that if this group commutes with the modular (KMS) automorphism group associated with a normal faithful state on M, 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 of M is completely characterized by elements in M. 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.