Applications of uniform convexity of noncommutative 𝐿^{𝑝}-spaces

Abstract

We consider noncommutative L p {L^p} -spaces, 1 > p > ∞ 1 > p > \infty , associated with a von Neumann algebra, which is not necessarily semifinite, and obtain some consequences of their uniform convexity. Among other results, we obtain (i) the norm continuity of the "absolute value part" map from each L p {L^p} -space onto its positive part; (ii) a certain continuity result on Radon-Nikodym derivatives in the context of positive cones introduced by H. Araki; and (iii) the necessary and sufficient condition for certain L p {L^p} -norm inequalities to become equalities. Some dominated convergence theorems for a probability gage are also considered.