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

TL;DR

The paper extends multiplier theory to left multipliers of Hilbert $C^*$-modules by unifying Bakic–Guljaš and Lance's categorical perspectives. It constructs $LM(V)=\mathrm{Hom}_A(A,V)$ with the embedding $\Gamma$, proving that $(LM(V),LM(A),\Gamma)$ is a maximal left strictly essential Banach extension of $V$ and establishing its universal properties across admissible triples. It investigates essential vs strictly essential extensions in both $C^*$- and Banach contexts, analyzes matrix-algebra amplifications with $LM(M_n(A))\simeq M_n(LM(A))$, and develops a left strict topology to provide a topological realization of left multipliers. The results unify algebraic and topological approaches to multipliers on Hilbert $C^*$-modules and lay groundwork for potential quasi-multiplier analogues in this setting, with implications for stabilization and module-frame theory in the non-unital regime.

Abstract

In the present paper we develop both ideas of D. Bakić and B. Gulja{š} 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.

Theorems & Definitions (45)

• Definition 2.1
• Definition 2.2
• Lemma 2.3
• Definition 2.4
• Definition 2.5
• Definition 2.6
• Theorem 2.7
• Theorem 2.8
• Definition 3.1
• Remark 3.2