Continuity and relative hamiltonians

Abstract

Let (ωn)n≧1 be a norm convergent sequence of normal states on a von Neumann algebraA withωn→ω. Let (kn)n≧1 be a strongly convergent sequence of self-adjoint elements ofA withkn→k. It is shown that the sequence\((\omega _n^{k_n } )_{n \geqq 1} \) of perturbed states converges in norm toωω. A related result holds forC*-algebras. A counter-example is provided to show that it is not sufficient to assume weak convergence of (ωn)n≧1 even whenkn=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.