Journal of Operator Theory

Volume 64, Issue 1, Summer 2010 pp. 149-154.

Continuity of CP-semigroups in the point-strong operator topology

Authors: Daniel Markiewicz (1) and Orr Moshe Shalit (2)
Author institution: (1) Department of Mathematics, University of Toronto, 40 St. George St., Room 6290, Toronto, ON M5S 2E4, Canada
(2) Department of Mathematics, Technion - Israel Institute of Technology, 32000, Haifa, Israel

Summary: We prove that if $\{\phi_t\}_{t \geqslant 0}$ is a CP-semigroup acting on a von Neumann algebra $M \subseteq B(H)$, then for every $A\in M$ and $\xi \in H$, the map $t \mapsto \phi_t(A)\xi$ is norm-continuous. We discuss the implications of this fact to the existence of dilations of CP-semigroups to semigroups of endomorphisms.

Keywords: CP-semigroup, E$_0$-semigroup, strong operator continuity, Bhat's dilation theorem, dilations.

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