States and automorphism groups of operator algebras

Richard H. Herman, Masamichi Takesaki

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)142-160
Number of pages19
JournalCommunications in Mathematical Physics
Issue number2
StatePublished - Jun 1970
Externally publishedYes

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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