### Abstract

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 language | English (US) |
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Pages (from-to) | 142-160 |

Number of pages | 19 |

Journal | Communications in Mathematical Physics |

Volume | 19 |

Issue number | 2 |

DOIs | |

State | Published - Jun 1 1970 |

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Physics and Astronomy(all)
- Mathematical Physics

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## Cite this

*Communications in Mathematical Physics*,

*19*(2), 142-160. https://doi.org/10.1007/BF01646631