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

TL;DR

This work studies when the zero functional on a Hilbert $A$-submodule $M$ has only the trivial extension to a larger module $N$ under the condition $M^\perp_N={0}$. It establishes, across $W^*$-algebras, monotone complete and compact C*-algebras, and for one-sided modular ideals, that such nontrivial modular extensions $r_0: N\to A$ do not exist and that the dual modules satisfy strong embeddings (e.g., $M'$ embeds into $N'$ as an orthogonal summand, often with $M'=N'$). The paper connects the nonexistence of $r_0$ to kernel properties of bounded modular operators, showing that kernels are biorthogonally complemented in these contexts and providing a new perspective on the structure of bounded modular operators. It also clarifies the validity of related lemmas in monotone complete and compact cases, while contrasting with counterexamples in general C*-algebras, thereby delineating where zero-extension phenomena occur and where they fail.

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.

Theorems & Definitions (37)

• Example 1.1
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Theorem 3.3
• proof
• Lemma 3.4
• proof
• Proposition 3.5