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

Watanabe, Seiji

doi:10.1090/S0002-9939-1988-0934853-X

$L_ p$-continuity of positive semigroups on finite von Neumann algebras
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by Seiji Watanabe

Proc. Amer. Math. Soc. 102 (1988), 840-842

DOI: https://doi.org/10.1090/S0002-9939-1988-0934853-X

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Abstract:

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

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