Multiplier modules of Hilbert $C^*$-modules revisited

Abstract: Following the approach to multiplier modules of Hilbert $C^*$-modules introduced by D. Bakić and B. Guljaš (2003) we reconsider key definitions and facts to get deeper insights into related structures. The independent approach by M. Daws (2010) and by A. Buss, B. Kwaśniewski, A. McKee, A. Skalski (2024) via Banach $C^*$-modules serves as an alternative point of view on which we comment and give facts to interrelate these two theories. 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 $C^*$-correspondence 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 for which continuation of these kinds of bounded module operators from the initial Hilbert $C^*$-module to its multiplier module fails are given, 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.

geometric topologynumber theoryoperator algebrasrepresentation theory

Audience: researchers in the topic

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