Localised Hilbert modules and weak noncommutative Cartan pairs

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 $C^{\ast }$-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.

Counterexamples to some of the main results have been found

References in corpus (4)

• Fell bundles over inverse semigroups and twisted etale groupoids
• Noncommutative Cartan sub-algebras of C*-algebras
• Aperiodicity: the almost extension property and uniqueness of pseudo-expectations
• Cartan subalgebras in C*-algebras