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Invariant states and conditional expectations of the anticommutation relations

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Abstract

The groupG of unitary elements of a maximal abelian von Neumann algebra on a separable, complex Hilbert spaceH acts as a group of automorphisms on the CAR algebraA(H) overH. It is shown that the set ofG-invariant states is a simplex, isomorphic to the set of regular probability measures on aw*-compact setS ofG-invariant generalized free states. The GNS Hilbert space induced by an arbitraryG-invariant state onA(H) supports a *-representation ofC(S); the canonical map ofA(H) intoC(S) can then be locally implemented by a normal,G-invariant conditional expectation.

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Authors and Affiliations

  1. Departments of Physics and of Mathematics, University of British Columbia, Vancouver, B. C., Canada

    John C. Wolfe

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  1. John C. Wolfe
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Communicated by H. Araki

Supported by National Research Council of Canada grants NRC A 3119 NRC A 6570, NRC A 126.

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Wolfe, J.C. Invariant states and conditional expectations of the anticommutation relations. Commun.Math. Phys. 44, 53–72 (1975). https://doi.org/10.1007/BF01609058

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  • DOI: https://doi.org/10.1007/BF01609058

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