Convergence in the dual of a σ-complete C∗-algebra

Let B be a monotone σ -complete C∗-algebra. Let (μn) (n = 1, 2, . . .) be a sequence in the dual of B such that limμn(p) exists for each projection p.We prove that the sequence must converge weakly. As an application, we obtain a non-commutative generalisation of the Brooks–Jewett Theorem.  2004 Elsevier Inc. All rights reserved.