On Essential and Topologically Essential Submodules of Hilbert C*-Modules

TL;DR

This paper investigates the concepts of essential and topologically essential submodules in Hilbert C*-modules, establishing their equivalence in various contexts and providing geometric insights in the commutative case.

Contribution

It proves the equivalence of essential and topologically essential submodules for right ideals in C*-algebras and extends this to Hilbert C*-modules, with a geometric reformulation in the commutative case.

Findings

Essential and topologically essential submodules are equivalent for right ideals in C*-algebras.

The equivalence extends to closed submodules of Hilbert C*-modules.

A fiberwise criterion for essentiality is derived in the commutative case.

Abstract

We study two notions of largeness for closed submodules of Hilbert C*-modules: essentiality and topological essentiality. While the analogous properties are known to be equivalent for closed two-sided ideals of C*-algebras, the one-sided case is more subtle. We prove that these two notions remain equivalent for closed right ideals of an arbitrary C*-algebra. Next, using the correspondence between submodules and right ideals of the algebra of compact operators, we extend this result to closed submodules of Hilbert C*-modules. In the commutative case, where a Hilbert module can be realized as a continuous field of Hilbert spaces, we give a geometric reformulation of essentiality and derive a fiberwise criterion.