Unbounded derivations and invariant states |
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Authors: | Ola Bratteli Uffe Haagerup |
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Affiliation: | (1) UER Scientifique de Marseille-Luminy, and Centre de Physique Theorique CNRS, F-13274 Marseille Cedex 2, France;(2) Matematisk Institut, Odense Universitet, DK-5230 Odense M, Denmark |
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Abstract: | Let be a von Neumann algebra with a cyclic and separating vector . Let =iH, ·] be the spatial derivation implemented by a selfadjoint operatorH, such thatH=0. Let be the modular operator associated with the pair (, ). We prove the equivalence of the following three conditions:1)H is essential selfadjoint onD(), andH commutes strongly with .2) The restriction ofH toD() is essential selfadjoint onD(1/2) equipped with the inner product(|)#=(|)+(1/2|1/2), , D(1/2).3) exp (itH) exp (–itH)= for anyt.We show by an example, that the first part of 1),H is essential selfadjoint onD(), does not imply 3). This disproves a conjecture due to Bratteli and Robinson 3].Part of this work was done while O.B. was a member of Zentrum für interdisziplinäre Forschung der Universität Bielefeld |
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