On Hahn-Banach type theorems for Hilbert C*-modules

TL;DR

This paper explores Hahn-Banach type extension theorems for bounded C*-linear maps on Hilbert C*-modules, linking these to properties of the underlying C*-algebras such as monotone completeness.

Contribution

It introduces three new extension criteria for bounded C*-linear maps and relates these to the structural properties of C*-algebras, including a characterization of (AW*)-C*-algebras.

Findings

Established three Hahn-Banach type extension criteria for Hilbert C*-modules.

Linked extension properties to monotone and additive completeness of C*-algebras.

Provided an alternative characterization of (AW*)-C*-algebras.

Abstract

We show three Hahn-Banach type extension criteria for (sets of) bounded C*-linear maps of Hilbert C*-modules to the underlying C*-algebras of coefficients. One criterion establishes an alternative description of the property of (AW*-) C*-algebras to be monotone complete or additively complete.