On Essential and Topologically Essential Submodules of Hilbert C*-Modules
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917400482414592 |
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| author | Kartvelishvili, Kirill |
| author_facet | Kartvelishvili, Kirill |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_10206 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On Essential and Topologically Essential Submodules of Hilbert C*-Modules Kartvelishvili, Kirill Operator Algebras 46L08 (Primary), 46L05, 46M20 (Secondary) 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. |
| title | On Essential and Topologically Essential Submodules of Hilbert C*-Modules |
| topic | Operator Algebras 46L08 (Primary), 46L05, 46M20 (Secondary) |
| url | https://arxiv.org/abs/2604.10206 |