𝐿_{𝑝}-continuity of positive semigroups on finite von Neumann algebras

Abstract

Let M M be a σ \sigma -finite, finite von Neumann algebra with a faithful normal tracial state τ \tau . Let α \alpha be a one-parameter semigroup of normal positive contractions of M M . Then it is shown that α \alpha is continuous with respect to the L p {L_p} -norm ( 1 ≤ p > ∞ ) (1 \leq p > \infty ) induced by τ \tau if and only if it is σ \sigma -weakly continuous.