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

Abstract

AbstractConsidering 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 $$M \subset N$$ M ⊂ N with $$M^\bot = \{ 0 \}$$ M ⊥ = { 0 } over a fixed C*-algebra A of coefficients cannot be separated by a non-trivial bounded A-linear functional $$r_0: N \rightarrow A$$ r 0 : N → 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 $$r_0$$ r 0 exist for a given pair of full Hilbert C*-modules $$M \subseteq N$$ M ⊆ N over a given C*-algebra A iff there exists a bounded A-linear non-adjointable operator $$T_0: N \rightarrow N$$ T 0 : N → N , such that the kernel of $$T_0$$ 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.