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

Michael Frank

TL;DR

This work addresses Hahn-Banach type extension problems for bounded $A$-linear maps on Hilbert $A$-modules and seeks criteria that preserve the $A$-codomain. It establishes an equivalence between two natural extension formulations (i) and (ii) and shows these imply that the multiplier algebra $M(A)$ is monotone complete (equivalently additively complete), which in turn yields the (iii)/(iv) implications. The authors develop a unified framework using $A^{**}$-duals and the associated ${\mathcal{M}}^\#$ construction to analyze when biorthogonal complements ${\mathcal{N}}^{\bot\bot}$ become orthogonal direct summands, and they provide explicit decomposition criteria ${\mathcal{M}}''={\mathcal{N}}^{\bot\bot}\oplus{\mathcal{N}}^\bot$ under suitable hypotheses. Placed in the broader context of operator-module Hahn-Banach theorems, monotone completeness emerges as a natural boundary for sharp codomain restrictions, with further connections to injective envelopes and the weak expectation property discussed as directions for future work.

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.

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

TL;DR

This work addresses Hahn-Banach type extension problems for bounded -linear maps on Hilbert -modules and seeks criteria that preserve the -codomain. It establishes an equivalence between two natural extension formulations (i) and (ii) and shows these imply that the multiplier algebra is monotone complete (equivalently additively complete), which in turn yields the (iii)/(iv) implications. The authors develop a unified framework using -duals and the associated construction to analyze when biorthogonal complements become orthogonal direct summands, and they provide explicit decomposition criteria under suitable hypotheses. Placed in the broader context of operator-module Hahn-Banach theorems, monotone completeness emerges as a natural boundary for sharp codomain restrictions, with further connections to injective envelopes and the weak expectation property discussed as directions for future work.

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.

Paper Structure

This paper contains 5 sections, 16 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The metric aspects of the proof
  4. The inner product aspects of the proof
  5. Final remarks

Key Result

Theorem 1.1

For C*-algebras $A$ the following conditions are equivalent: (i) $\,\,$ for any pair $\{ \{ {\mathcal{M}}, \langle .,. \rangle \}, {\mathcal{N}} \subseteq {\mathcal{M}} \}$ of a Hilbert $A$-module and a Hilbert $A$-sub-mo-dule there exists an $A$-linear isometric embedding $\phi$ of the Banach $A$-m

Theorems & Definitions (27)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Lemma 2.4
  • proof
  • Lemma 3.1
  • proof
  • ...and 17 more