Dilation theory for right LCM semigroup dynamical systems

TL;DR

This paper develops a dilation theory for right LCM semigroup actions on C*-algebras, generalizing previous results and providing conditions for dilating covariant representations to isometric ones.

Contribution

It introduces a generalized Stinespring dilation theorem for right LCM semigroup dynamical systems and establishes criteria for dilating covariant representations to isometric boundary quotient representations.

Findings

Proves a generalized Stinespring's dilation theorem for these systems.

Provides sufficient conditions for dilating covariant representations to isometric representations.

Extends earlier dilation results to a broader class of semigroup actions.

Abstract

This paper examines actions of right LCM semigroups by endomorphisms of C*-algebras that encode an additional structure of the right LCM semigroup. We define contractive covariant representations for these semigroup dynamical systems and prove a generalized Stinespring's dilation theorem showing that these representations can be dilated if and only if the map on the C*-algebra is unital and completely positive. This generalizes earlier results about dilations of right LCM semigroups of contractions. In addition, we also give sufficient conditions under which a contractive covariant representation of a right LCM system can be dilated to an isometric representation of the boundary quotient.