A bounded sequence of normal functionals has a subsequence which is nearly weakly convergent

Let (φn) be a norm bounded sequence in the pre-dual of a von Neumann algebra M. In general it is not true that this sequence has a weakly convergent subsequence. But given a normal state ψ, then, for any ε > 0, there exists a projection e such that ψ(1 − e) ε and the restriction of (φn) to eMe has a subsequence which converges weakly to a normal functional on eMe.  2002 Elsevier Science (USA). All rights reserved