On the extendability of some classes of maps on Hilbert $C^*$-modules

TL;DR

This paper studies the extendability of certain classes of maps on Hilbert $C^*$-modules, showing conditions for extensions, providing counterexamples, and exploring categorical properties related to injective objects.

Contribution

It establishes that every completely semi-$$-map on a submodule extends to the entire module and characterizes injective objects in a related category.

Findings

Every completely semi-$$-map on a submodule extends to the whole module.

Some $$-maps cannot be extended beyond their initial domain.

Identification of injective objects in the category of Hilbert $C^*$-modules.

Abstract

In this paper, we show that every completely semi-$\phi$-map on a submodule of a Hilbert $C^*$-module has a completely semi-$\phi$-map extension on the whole of module. We also investigate the extendability of $\phi$-maps and provide examples of $\phi$-maps which has no $\phi$-map extension. Finally, we introduce a category of Hilbert $C^*$-module and determine injective objects in this category.