Dilations of interaction groups that extend actions of Ore semigroups

TL;DR

This paper proves that interaction groups extending Ore semigroup actions on C*-algebras can be dilated to group actions on larger C*-algebras, with uniqueness and faithfulness conditions under certain assumptions.

Contribution

It establishes a dilation theorem for interaction groups extending Ore semigroup actions, generalizing previous results and providing conditions for faithfulness and minimality.

Findings

Every interaction group admits a dilation to a group action.

The dilation is unique under a natural minimality condition.

Faithfulness of the expectation is guaranteed when induced by covering maps.

Abstract

We show that every interaction group extending an action of an Ore semigroup by injective unital endomorphisms of a C*-algebra, admits a dilation to an action of the corresponding enveloping group on another unital C*-algebra, of which the former is a C*-subalgebra: the interaction group is obtained by composing the action with a conditional expectation. The dilation is essentially unique if a certain natural condition of minimality is imposed. If the action is induced by covering maps on the spectrum, then the expectation is faithful.