Abstract
Quite recently a criterion for the \(\mathcal{A}\)-compactness of an ajointable operator \(F\colon {\mathcal M} \to\mathcal{N}\) between Hilbert \(C^*\)-modules, where \(\mathcal{N}\) is countably generated, was obtained. Namely, a uniform structure (a system of pseudometrics) in \(\mathcal{N}\) was discovered such that \(F\) is \(\mathcal{A}\)-compact if and only if \(F(B)\) is totally bounded, where \(B\subset {\mathcal M} \) is the unit ball.
We prove that (1) for a general \(\mathcal{N}\), \(\mathcal{A}\)-compactness implies total boundedness, (2) for \(\mathcal{N}\) with \(\mathcal{N}\oplus K\cong L\), where \(L\) is an uncountably generated \(\ell_2\)-type module, total boundedness implies compactness, and (3) for \(\mathcal{N}\) close to be countably generated, it suffices to use only pseudometrics of “frame-like origin” to obtain a criterion for \(\mathcal{A}\)-compactness.
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Acknowledgments
The authors are grateful to A. I. Korchagin for helpful discussions.
Funding
This work was supported by the Theoretical Physics and Mathematics Advancement Foundation “BASIS.”
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Translated from Funktsional'nyi Analiz i ego Prilozheniya, 2020, Vol. 54, pp. 74-84 https://doi.org/10.4213/faa3809.
Translated by E. V. Troitsky and D. V. Fufaev
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Troitsky, E.V., Fufaev, D.V. Compact Operators and Uniform Structures in Hilbert \(C^*\)-Modules. Funct Anal Its Appl 54, 287–294 (2020). https://doi.org/10.1134/S0016266320040061
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DOI: https://doi.org/10.1134/S0016266320040061