On large submodules in Hilbert C⁎-modules

We consider several natural ways of expressing the idea that a one-sided ideal in a C⁎-algebra (or a submodule in a Hilbert C⁎-module) is large, and show that they differ, unlike the case of two-sided ideals in C⁎-algebras. We then show how these different notions, for ideals and for submodules, are...

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Bibliographic Details
Published in:Journal of mathematical analysis and applications 2025-02, Vol.542 (2), p.128781, Article 128781
Main Author: Manuilov, V.
Format: Article
Language:English
Subjects:
[formula omitted]-algebra
Hilbert [formula omitted]-module
Right ideal
Submodule
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Summary:We consider several natural ways of expressing the idea that a one-sided ideal in a C⁎-algebra (or a submodule in a Hilbert C⁎-module) is large, and show that they differ, unlike the case of two-sided ideals in C⁎-algebras. We then show how these different notions, for ideals and for submodules, are related. We also study some permanence properties for these notions. Finally, we use essential right ideals to extend the inner product on a Hilbert C⁎-module to a part of the dual module.
ISSN:0022-247X
DOI:10.1016/j.jmaa.2024.128781