On a weakly convergent sequence of normal functionals on a von Neumann algebra

Abstract

G. F. Dell'Antonio [1] recently has discussed weakly convergent sequences of normal states of von Neumann algebras and proved that every weakly convergent sequence of normal states of a factor of type I converges also uniformly. Moreover, he has shown that this statement is not true for a factor of type II. The purpose of this note is to investigate when a weakly convergent sequence of normal states converges also uniformly in the case of type II factors. We shall confine ourselves to the class of normal generalized irreducible functionals on a factor of type II. Then the generalized irreducibility of functionals makes it possible to find a simple and relevant condition for our problem. Throughout this paper, for convenience functional will always mean a positive linear functional on a von Neumann algebra. Let us recall that a functional p on a von Neumann algebra M is said to be generalized irreducible on M if whenever X is a functional on M such that that Xo ?Xp for some positive constant X (i.e., w(A) :5Xp(A) for all positive operators A in M), there exists a positive operator B in M such that w (A) =p(AB) for all A ?M. As is well known, every normal trace of a finite von Neumann algebra is generalized irreducible (see [4, Lemma 14.1]). We say that a sequence {pn} of functionals on a von Neumann algebra M is bounded from below by a functional p on M if p <Pn for all n. Then we shall prove the following.