Peripheral Poisson Boundary: Extensions and Examples

TL;DR

This paper extends the concept of peripheral Poisson boundary from unital completely positive maps to broader classes of maps and semigroups on von Neumann algebras, revealing new properties and challenges.

Contribution

It introduces an extension of peripheral Poisson boundary theory to contractive maps and semigroups on von Neumann algebras, highlighting unitality and computational formulas.

Findings

Peripheral Poisson boundary is unital when nontrivial.

Extension theory works well for von Neumann algebras and normal maps.

Obstacles exist in the $C^*$-algebra framework.

Abstract

The main purpose of this article is to explore the possibility of extending the notion of peripheral Poisson boundary of unital completely positive (UCP) maps to contractive completely positive (CCP) maps and to unital and non-unital contractive quantum dynamical semigroups on von Neumann algebras. We observe that the theory extends easily in the setting of von Neumann algebras and normal maps. Surprisingly, the peripheral Poisson boundary is unital, whenever it is nontrivial, even for contractive semigroups. The strong operator limit formula for computing the extended Choi-Effros product remains intact. However, there are serious obstacles in the framework of $C^*$-algebras, and we are unable to define the extended Choi-Effros product in such generality. We provide several intriguing examples to illustrate this.