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arXiv · 2604.10206

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

Kirill Kartvelishvili
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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.

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Kirill Kartvelishvili. 2026-04-11. On Essential and Topologically Essential Submodules of Hilbert C*-Modules. https://arxiv.org/abs/2604.10206

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