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Unbounded derivations and invariant states

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Abstract

Let ℳ be a von Neumann algebra with a cyclic and separating vector Ω. Let δ=i[H, ·] be the spatial derivation implemented by a selfadjoint operatorH, such thatHΩ=0. Let Δ be the modular operator associated with the pair (ℳ, Ω). We prove the equivalence of the following three conditions:

1)H is essential selfadjoint onD(δ)Ω, andH commutes strongly with Δ.

2) The restriction ofH toD(δ)Ω is essential selfadjoint onD(Δ1/2) equipped with the inner product

(ξ|η)#=(ξ|η)+(Δ1/2ξ|Δ1/2η), ξ, η ∈D(Δ1/2).

3) exp (itH) ℳ exp (−itH)=ℳ for anyt∈ℝ.

We show by an example, that the first part of 1),H is essential selfadjoint onD(δ)Ω, does not imply 3). This disproves a conjecture due to Bratteli and Robinson [3].

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Authors and Affiliations

  1. UER Scientifique de Marseille-Luminy, and Centre de Physique Theorique CNRS, F-13274, Marseille Cedex 2, France

    Ola Bratteli

  2. Matematisk Institut, Odense Universitet, DK-5230, Odense M, Denmark

    Uffe Haagerup

Authors
  1. Ola Bratteli
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  2. Uffe Haagerup
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Additional information

Communicated by H. Araki

Part of this work was done while O.B. was a member of Zentrum für interdisziplinäre Forschung der Universität Bielefeld

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Bratteli, O., Haagerup, U. Unbounded derivations and invariant states. Commun.Math. Phys. 59, 79–95 (1978). https://doi.org/10.1007/BF01614156

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  • DOI: https://doi.org/10.1007/BF01614156

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