Construction of enveloping actions

TL;DR

This paper investigates the conditions under which partial actions on *-algebras, C*-algebras, and Hilbert modules can be globalized, providing necessary and sufficient criteria especially for C*-algebras.

Contribution

It establishes a necessary and sufficient condition for the existence of a globalization of partial actions on C*-algebras and extends this to Hilbert modules via linking algebra.

Findings

Necessary condition for globalization on *-algebras

Necessary and sufficient condition for C*-algebras

Translation of conditions to Hilbert modules

Abstract

We study the problem of constructing a globalization for partial actions on *-algebras, C*-algebras and Hilbert modules. For the first ones we give a necessary condition for the existence of a globalization and we prove this conditions is necessary and sufficient for C*-algebras. Using the linking algebra of a Hilbert module we translate this condition to the realm of partial action on Hilbert modules.