Absolute Continuity in Noncommutative Measure Theory

Recent results on absolute continuity of Banach space valued operators and convergence theorems on operator algebras are deepened and summarized It is shown that absolute continuity of an operator T on a von Neumann algebra M with respect to a positive normal functional psi on M is not implied by the fact that the null projections of psi are the null projections of T However, it is proved that the implication above is true whenever M is finite or T is weak*-continuous Further it is shown that the absolute value preserves the Vitali-Hahn-Saks property if, and only if, the underlying algebra is finite This result Improves classical results on weak compactness of sets of noncommutative measures