Multiplier modules of Hilbert C*-modules revisited
Michael Frank
TL;DR
The paper revisits multiplier modules $M(X)$ of full Hilbert $A$-modules to obtain finer properties of these extensions and their operator structures. It shows that being a multiplier module is invariant under viewing the same module from the left or right Morita-equivalent perspective, and it establishes intrinsic links between ${ m K}_A(X)$, ${ m End}_A(X)$, and their multiplier counterparts over $M(A)$, including strict-density and strict completion aspects. The work demonstrates that while ${ m K}_A(X)$ embeds into ${ m K}_{M(A)}(M(X))$ and ${ m End}_{M(A)}(M(X))$ embeds into ${ m End}_A(X)$, these embeddings are generally proper and not every operator on $X$ has a norm-preserving continuation to $M(X)$; when continuations exist they are unique. It also shows limitations of Hahn–Banach-type extension results for bounded modular functionals in this setting, highlighting subtle interplays between extension theory, Morita equivalence, and multiplier structures in Hilbert C*-modules.
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.