A study of super-KMS functionals

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1989
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Virginia Polytechnic Institute and State University
Abstract

We study properties of super-KMS functionals on ℤ₂ graded von Neumann algebras. We prove that if a normal self-adjoint functional ω is weakly super-KMS, then the uniquely defined by the polar decomposition of ω positive functional |ω| is KMS.

We construct a graded representation of any von Neumann algebra with a normal self-adjoint super-KMS functional on it as an algebra of bounded operators on a Hilbert space. The grading of the algebra of operators that we obtain is induced from a natural orthogonal decomposition of the Hilbert space. In our construction we have to use the weak super-KMS property and the implications we have derived from it.

We present a generalization of the Tomita — Takesaki theorem to the case of (not necessarily positive) self-adjoint normal faithful functionals. We show that for every such functional ω there is a canonically defined *-automorphism group (the analog of the modular group) and a canonical ℤ₂ grading of the algebra, commuting with the automorphism group. The functional ω is weakly super-KMS with respect to them. Furthermore, the canonical automorphism group and ℤ₂ grading are the unique pair of a σ-weakly continuous one-parameter *-automorphism group and a ℤ₂ grading, commuting with each other, with respect to which ω is super-KMS.

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