Unital Dilations of Completely Positive Semigroups

TL;DR

This paper advances the theory of dilating completely positive semigroups to unital endomorphism semigroups, focusing on constructing continuous unital dilations and linking them with free probability.

Contribution

It extends Sauvageot's dilation theory to ensure the existence of continuous unital dilations while preserving semigroup continuity properties.

Findings

Established existence of continuous unital dilations

Connected dilation theory with free probability

Improved understanding of semigroup embeddings

Abstract

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.