On bounded module functionals and operators on Hilbert C*-modules without biorthogonally complemented kernels

Abstract

We investigate the structural reasons behind the existence and non-existence of bounded module functionals and bounded module operators on Hilbert C*-modules whose kernels are not orthogonally complemented. The key motivating example was introduced by J. Kaad and M. Skeide in their 2023 paper. Building on their ideas and related developments, we obtain a structural characterization of this class of examples in a more abstract theoretical framework. We also study non-adjointable bounded module operators associated with such module functionals, together with the properties of the corresponding operator algebras and operator spaces. Our results provide a deeper understanding of phenomena beyond the adjointable setting, place several existing partial results into a unified framework, and show how large families of further examples can be constructed systematically.

17 pages. This is a completely revised version with an analysis of the Kaad-Skeide counterexample, with results on its theoretical background and with consequences. Thank you for communications and discussions