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

TL;DR

This paper investigates the structural reasons behind the behavior of bounded A-linear maps on Hilbert C*-modules, focusing on the self-adjoint case and highlighting open problems in the general case.

Contribution

It provides initial structural insights into bounded module maps vanishing on submodules, especially in the self-adjoint setting, and discusses limitations in existing proofs.

Findings

Self-adjoint case proved; general case remains open.

Bounded A-linear maps vanishing on certain submodules are only zero.

Identifies a flaw in previous proof regarding operator commutation.

Abstract

We find first structural background information about the reasons that for any C*-algebra $A$ and any two Hilbert $A$-modules $M \subseteq N$ with $M^\perp=\{0\}$, every bounded $A$-linear map $N \to A$ (or $N \to N)$ vanishing on $M$ might be only the zero map. The self-adjoint case is proved, whereas the general case is open with partial insights. Unfortunately, the proof of Lemma 3.3 of our first version contains the implicit assumption that the projection $P$ and the operator $S$ commute, which is not the case for non-zero non-self-adjoint operators $S$.