Localised Hilbert modules and weak noncommutative Cartan pairs

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.