Regularity results for classes of Hilbert C*-modules with respect to special bounded modular functionals

TL;DR

This paper investigates the conditions under which bounded A-linear functionals uniquely extend from a submodule to a Hilbert C*-module, revealing new insights into modular operators and their properties.

Contribution

It establishes the uniqueness of bounded A-linear functional extensions in specific C*-algebra contexts and links this to properties of modular operators.

Findings

Uniqueness of extension holds over W*-algebras, monotone complete, and compact C*-algebras.

Existence of non-zero separating functionals relates to non-adjointable operators with non-biorthogonally closed kernels.

Provides a corrected proof of a key lemma in certain C*-algebra cases.

Abstract

Considering the deeper reasons of the appearance of a remarkable counterexample by J.~Kaad and M.~Skeide [17] we consider situations in which two Hilbert C*-modules $M \subset N$ with $M^\bot = \{ 0 \}$ over a fixed C*-algebra $A$ of coefficients cannot be separated by a non-trivial bounded $A$-linear functional $r_0: N \to A$ vanishing on $M$. In other words, the uniqueness of extensions of the zero functional from $M$ to $N$ is focussed. We show this uniqueness of extension for any such pairs of Hilbert C*-modules over W*-algebras, over monotone complete C*-algebras and over compact C*-algebras. Moreover, uniqueness of extension takes place also for any one-sided maximal modular ideal of any C*-algebra. Such a non-zero separating bounded $A$-linear functional $r_0$ exist for a given pair of full Hilbert C*-modules $M \subseteq N$ over a given C*-algebra $A$ iff there exists a bounded $A$-linear non-adjointable operator $T_0: N \to N$ such that the kernel of $T_0$ is not biorthogonally closed w.r.t. $N$ and contains $M$. This is a new perspective on properties of bounded modular operators that might appear in Hilbert C*-module theory. By the way, we find a correct proof of [13, Lemma 2.4] in the case of monotone complete and compact C*-algebras, but not in the general C*-case.