Strict essential extensions of C*-algebras and Hilbert C*-modules

TL;DR

This paper advances the theory of multipliers for Hilbert C*-modules by exploring essential extensions, maximality properties, and topological approaches, connecting categorical and algebraic perspectives.

Contribution

It introduces new results on maximality and relations of essential extensions for Hilbert C*-modules, integrating categorical and topological methods.

Findings

Maximality of strictly essential extensions established.

Relations between essential and strictly essential extensions clarified.

Topological approach to left multiplier theory developed.

Abstract

In the present paper we develop both ideas of D. Baki\'c and B. Gulja{\v{s}} and the categorical approach to multipliers from E.C. Lance's book and publications of the second author, for the introduction and study of left multipliers of Hilbert $C^*$-modules. Some properties and, in particular, the property of maximality among all strictly essential extensions of a Hilbert $C^*$-module for left multipliers are proved. Also relations between left essential and left strictly essential extensions in different contexts are obtained. Left essential and left strictly essential extensions of matrix algebras are considered. In the final paragraph the topological approach to the left multiplier theory of Hilbert $C^*$-modules is worked out.