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Continuity and relative hamiltonians

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Abstract

Let (ω n ) n ≧1 be a norm convergent sequence of normal states on a von Neumann algebraA withω n →ω. Let (k n ) n≧1 be a strongly convergent sequence of self-adjoint elements ofA withk n →k. It is shown that the sequence\((\omega _n^{k_n } )_{n \geqq 1} \) of perturbed states converges in norm toω ω. A related result holds forC *-algebras. A counter-example is provided to show that it is not sufficient to assume weak convergence of (ω n ) n ≧1 even whenk n=k for alln. However, conditions are given which, together with weak convergence, are sufficient. Relative entropy methods are used, and a relative entropy inequality is proved.

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  1. The Cavendish Laboratory, Madingley Road, CB3 0HE, Cambridge, UK

    Matthew J. Donald

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  1. Matthew J. Donald
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Communicated by H. Araki

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Donald, M.J. Continuity and relative hamiltonians. Commun.Math. Phys. 136, 625–632 (1991). https://doi.org/10.1007/BF02099078

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  • DOI: https://doi.org/10.1007/BF02099078

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