Localised Hilbert modules and weak noncommutative Cartan pairs

TL;DR

This paper introduces the concept of localisation for Hilbert modules, enabling the extension of non-closed actions on C*-algebras to closed actions on local multiplier algebras, and characterizes weak Cartan inclusions as inverse semigroup crossed products.

Contribution

It defines weak Cartan inclusions and characterizes them as crossed products by inverse semigroup actions, generalizing Renault's results to a broader setting.

Findings

Weak Cartan subalgebras are maximal abelian in the commutative case.

Localisation of Hilbert modules enriches the analysis of C*-algebra actions.

Known results on closed actions are extended to unclosed actions via localisation.

Abstract

We define the localisation of a Hilbert module in analogy to the local multiplier algebra. We use properties of this localisation to enrich non-closed actions on $C^*$-algebras to closed actions on local multiplier algebras, and descend known results on such closed actions down to their unclosed counterparts. We define weak Cartan inclusions and characterise them as crossed products by inverse semigroup actions. We show that in the commutative case we show that weak Cartan subalgebras are maximal abelian, thereby generalising the case studied by Renault.