arXiv subjects
Publications and source records attributed to David J. Gaebler.
Using ideas due to Jean-Luc Sauvageot, we prove the existence of a continuous unital dilation of a CP$_0$-semigroup on a separable W$^*$-algebra. This paper presents the material in the author's Ph. D. thesis (arXiv.org:1304.0134.pdf) with some generalizations and an improved exposition.
Dilations of completely positive semigroups to endomorphism semigroups have been studied by numerous authors. Most existing dilation theorems involve a non-unital embedding, corresponding to the embedding of $B(H)$ as a corner of $B(K)$ for Hilbert spaces $H \subset K$. A 1986 paper of Jean-Luc Sauvageot shows how to achieve a unital dilation, but does not specify how to do so while also preserving continuity properties of the original semigroup. This thesis further develops Sauvageot's dilation theory in order to establish the existence of continuous unital dilations, and to explore connections with free probability.