Product systems, subproduct systems and dilation theory of completely positive semigroups

TL;DR

This thesis develops a dilation theory for semigroups of completely positive maps, introducing subproduct systems and exploring their applications in multivariable operator theory and operator algebras.

Contribution

It introduces subproduct systems as a key technical tool for dilation theory of general semigroup-parameterized completely positive maps, extending existing frameworks.

Findings

Dilation theory for two-parameter semigroups developed

Subproduct systems characterized and linked to operator algebras

Applications to multivariable operator theory demonstrated

Abstract

This thesis is dedicated to developing a dilation theory for semigroups of completely positive maps. The first part treats two-parameter semigroups, and contains also contributions to dilation theory of product system representations. The second part deals with completely positive semigroups parameterized by quite general semigroups, where the major technical tool introduced is subproduct systems and their representations. In the third part subproduct systems are studied, together with the multivariable operator theory and operator algebras they give rise to.