Absolute Continuity in Noncommutative Measure Theory

ISSN： 0020-7748

Source： International Journal of Theoretical Physics, Vol.49, Iss.12, 2010-12, pp. : 3139-3145

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Abstract

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 ψ on M is not implied by the fact that the null projections of ψ 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.

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