The Quarterly Journal of Mathematics Advance Access published on June 7, 2006 Quart. J. Math. 57 (2006), 479–493; doi:10.1093/qmath/hal006 VITALI–HAHN–SAKS THEOREM FOR VECTOR MEASURES ON OPERATOR ALGEBRAS by EMMANUEL CHETCUTI (Department of Mathematics, Junior College, University of Malta) and JAN HAMHALTER (Department of Mathematics, Faculty of Engineering,Technicka 2, Prague 6, 16627, Czech Republic) [Received 30 August 2005. Revised 13 February 2006] Abstract We show that the direct generalization of theVitali–Hahn–Saks theorem is not valid for all measures on von Neumann algebras. By applying a general equicontinuity argument, we prove a direct extension of the Vitali–Hahn–Saks theorem for a wide range of vector measures on von Neumann algebras and JBW algebras. We also characterize relatively compact sets of vector measures on operator algebras. 1. Introduction The Vitali–Hahn–Saks theorem is one of the fundamental results in measure theory that can be presented in various forms. There are many different proofs of this theorem relating it to rather distinct concepts of measure theory [6–8, 18, 21]. These aspects, as well as the applications of this theorem to measure, and probability theories, have attracted many researchers. Our paper is devoted to the non-commutative analogy of the following version of the Vitali–Hahn–Saks theorem for completely additive measures (see for example [18, 21]). THEOREM 1.1 (Vitali–Hahn–Saks theorem) Let A be a complete algebra of subsets of a set  and ca(A) the set of all (complex) completely additive measures on A endowed with the topology of pointwise convergence on elements of A. Suppose that K ⊂ ca(A) is a relatively compact set such that each element of K is absolutely continuous with respect to a positive completely additive measure µ on A. Then K is uniformly absolutely continuous with respect to µ. The first generalization of the Vitali–Hahn–Saks theorem to von Neumann algebras was given by Aarnes [1] and Akemann [2]. Using deep results on the topological structure of preduals of von Neumann algebras, Akemann proved that a weakly relatively compact set K in the predual of a von Neumann algebra M is uniformly absolutely continuous with respect to a normal state ψ on M whenever the absolute value of each functional in K is absolutely continuous with respect to ψ . In particular, this establishes a direct generalization of the Vitali–Hahn–Saks theorem (Theorem 1.1) for the case of positive normal functionals. In the recent papers by Brooks et al. [11] and Brooks and Wright [9], similar results have been obtained for weakly relatively compact sets in the duals of general C ∗ -algebras. This interesting development made it possible to prove the Vitali–Hahn–Saks theorem for pointwise convergent sequences of finitely additive states on von Neumann algebras [9]. 479 © 2006. Published by Oxford University Press. All rights reserved For permissions, please email: journals.permissions@oxfordjournals.org