Dominated convergence theorems in Haagerup noncommutative Lp -spaces

Abstract

Let M M be a σ σ-finite von Neumann algebra and T:M→M T:M→M be a linear bounded positive map under some natural conditions. We obtain that if (x n)n≥1(x n) n≥1 is a sequence in M converging to x almost uniformly and (x n)n≥1(x n ) n≥1 satisfies certain domination condition, then (T(xn))n≥1(T(x n )) n≥1 converges to T(x) almost uniformly.