The intersection of complemented submodules is not complemented

TL;DR

This paper demonstrates through counterexamples that the intersection of complemented submodules in a Hilbert C*-module may not be complemented, resolving an open question in the field.

Contribution

It provides the first known counterexamples showing that intersections of complemented submodules need not be complemented in Hilbert C*-modules.

Findings

Counterexamples show non-complemented intersections

Answers an open question in the theory of Hilbert C*-modules

Highlights limitations of submodule complementarity

Abstract

We show by (counter)example that the intersection of complemented submodules in a Hilbert $C^*$-module is not necessarily complemented, answering an open question from [MR].