Beppo levi and Lebesgue type theorems for bundle convergence in noncommutative L 2-spaces

Abstract

Our subject is noncommutative measure theory in the context of von Neumann algebras. We consider a von Neumann algebra 21 equipped with a faithful normal state. The notion of bundle convergence for sequences of operators in 21. as well as for sequences of vectors in its L 2-completion were introduced in 1996 by Hensz, Jajte and Paszkiewicz [2], and this notion turned out to be an appropriate substitute for almost everywhere convergence in the noncommutative setting. In terms of bundle convergence, we prove the noncommutative versions of the standard Beppo Levi Theorem (for an increasing sequence of operators), the Lebesgue Dominated Convergence Theorem, in particular, the Arzela—Osgood Theorem (for a bounded sequence of operators). Only classical results of von Neumann algebras are required in our proofs. As an auxiliary result, we present a proof for the existence of a bundle which is contained in each of a countable number of given bundles in 21.