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

Abstract

Considering the deeper reasons of the appearance of a remarkable counterexample by Kaad and Skeide (J Operat Theory 89(2):343–348, 2023) 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 \(r_0: N \rightarrow A\) 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 \(T_0: N \rightarrow N\), 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 Lemma 2.4 of Frank (Int J Math 13:1–19, 2002) in the case of monotone complete and compact C*-algebras, but find it not valid in certain particular cases.