Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Bounded modular functionals and operators on Hilbert C*-modules are regular

Michael Frank, Cristian Ivanescu

Abstract

We prove that for any C*-algebra $A$ and Hilbert $A$-modules $M\subseteq N$ with $M^\perp=\{0\}$, every bounded $A$-linear map $N\to A$ (or $N\to N)$ vanishing on $M$ is the zero map. This verifies the conjectures of the first author and settles the regularity problem for bounded modular functionals and operators on Hilbert C*-modules. As a consequence, kernels of bounded C*-linear operators on Hilbert C*-modules are shown to be biorthogonally complemented, which gives a correct proof of Lemma 2.4 in ``On Hahn-Banach type theorems for Hilbert C*-modules'', Internat. J. Math. 13(2002), 1--19, in full generality.

Bounded modular functionals and operators on Hilbert C*-modules are regular

Abstract

We prove that for any C*-algebra and Hilbert -modules with , every bounded -linear map (or vanishing on is the zero map. This verifies the conjectures of the first author and settles the regularity problem for bounded modular functionals and operators on Hilbert C*-modules. As a consequence, kernels of bounded C*-linear operators on Hilbert C*-modules are shown to be biorthogonally complemented, which gives a correct proof of Lemma 2.4 in ``On Hahn-Banach type theorems for Hilbert C*-modules'', Internat. J. Math. 13(2002), 1--19, in full generality.
Paper Structure (3 sections, 5 theorems)

This paper contains 3 sections, 5 theorems.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. On regularity of bounded modular functionals and operators

Key Result

Lemma 3.2

(Cf. Frank_2024.) Let $A$ be a W*-algebra. Let $\mathcal{M}$ be a Hilbert $A$-submodule of a Hilbert $A$-module $\mathcal{N}$ such that the orthogonal complement of $\mathcal{M}$ relative to $\mathcal{N}$ is trivial, i.e. equals $\{0\}$. Then there does not exist any non-trivial bounded $A$-linear m

Theorems & Definitions (12)

  • Conjecture 1.1
  • Conjecture 1.2
  • Example 3.1
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Theorem 3.5
  • Remark 3.6
  • ...and 2 more