Decomposition of operator semigroups on W*-algebras

TL;DR

This paper establishes a decomposition for operator semigroups on W*-algebras, separating stable and reversible components, and advances the spectral theory for positive operators in this setting.

Contribution

It introduces a Jacobs-DeLeeuw-Glicksberg type decomposition for semigroups on W*-algebras, providing new structural insights and spectral analysis tools.

Findings

Decomposition into stable and reversible parts

Application to Perron-Frobenius spectral theory

Structural framework for positive operators

Abstract

We consider semigroups of operators on a W$^*$-algebra and prove, under appropriate assumptions, the existence of a Jacobs-DeLeeuw-Glicksberg type decomposition. This decomposition splits the algebra into a "stable" and "reversible" part with respect to the semigroup and yields, among others, a structural approach to the Perron-Frobenius spectral theory for completely positive operators on W$^*$-algebras.