Multiplier modules of Hilbert C*-modules revisited

TL;DR

This paper revisits the theory of multiplier modules of Hilbert C*-modules, establishing new properties, invariance under Morita equivalence, and characterizing relationships between various operator algebras and modules.

Contribution

It introduces new invariance properties of multiplier modules, characterizes their relationships with operator algebras, and analyzes the uniqueness of bounded module operator continuations.

Findings

Multiplier module property is invariant under Morita equivalence.

Bounded module operator continuations are always unique.

Examples show some continuations may fail, but are uniquely determined when they exist.

Abstract

The theory of multiplier modules of Hilbert C*-modules is reconsidered to obtain more properties of these special Hilbert C*-modules. The property of a Hilbert C*-module to be a multiplier C*-module is shown to be an invariant with respect to the consideration as a left or right Hilbert C*-module in the sense of a imprimitivity bimodule in strong Morita equivalence theory. The interrelation of the C*-algebras of ''compact'' operators, the Banach algebras of bounded module operators and the Banach spaces of bounded module operators of a Hilbert C*-module to its C*-dual Banach C*-module, are characterized for pairs of Hilbert C*-modules and their respective multiplier modules. The structures on the latter are always isometrically embedded into the respective structures on the former. Examples are given for which continuation of these kinds of bounded module operators from the initial Hilbert C*-module to its multiplier module fails. However, existing continuations turn out to be always unique. Similarly, bounded modular functionals from both kinds of Hilbert C*-modules to their respective C*-algebras of coefficients are compared, and eventually existing continuations are shown to be unique.