On the amenability of partial and enveloping actions

TL;DR

This paper establishes that the amenability of a partial action is equivalent to that of its Morita enveloping action, with applications to partial representations and invariant states in discrete groups.

Contribution

It proves the equivalence of amenability between partial actions and their Morita enveloping actions, extending results to partial representations and invariant states.

Findings

Partial actions are amenable iff their Morita enveloping actions are amenable.

Any partial representation of a discrete group is positive definite.

Extends Zeller-Meier's result on group amenability and invariant states to partial actions.

Abstract

We prove that a partial action is amenable if and only if so is its Morita enveloping action. As applications we prove that any partial representation of a discrete group is positive definite, and we extend a result of Zeller-Meier concerning the amenability of discrete groups and the existence of invariant states to partial actions.