CONTINUITY AND RELATIVE HAMILTONIANS

Let (omega-n)n greater-than-or-equal-to 1 be a norm convergent sequence of normal states on a von Neumann algebra A with omega-n --> omega. Let (k(n))n greater-than-or-equal-to 1 be a strongly convergent sequence of self-adjoint elements of A with k(n) --> k. It is shown that the sequence (omega-n(k)n)n greater-than-or-equal-to 1 of perturbed states converges in norm to omega-k. A related result holds for C*-algebras. A counter-example is provided to show that it is not sufficient to assume weak convergence of (omega-n)n greater-than-or-equal-to 1 even when k(n) = k for all n. However, conditions are given which, together with weak convergence, are sufficient. Relative entropy methods are used, and a relative entropy inequality is proved.