Abstract
We construct an example of a Hilbert \(C^*\)-module which shows that Troitsky’s theorem on the geometric essence of \( {\mathcal A} \)-compact operators between Hilbert \(C^*\)-modules cannot be extended to modules which are not countably generated case (even in the case of a stronger uniform structure, which is also introduced). In addition, the constructed module admits no frames.
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Acknowledgments
Author is grateful to E. V. Troitsky, K. L. Kozlov, and A. I. Korchagin for helpful discussions.
Funding
The work was supported by the Theoretical Physics and Mathematics Advancement Foundation “BASIS.”
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Translated from Funktsional'nyi Analiz i ego Prilozheniya, 2022, Vol. 56, pp. 94–105 https://doi.org/10.4213/faa3921.
Translated by D. V. Fufaev
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Fufaev, D.V. A Hilbert \(C^*\)-Module with Extremal Properties. Funct Anal Its Appl 56, 72–80 (2022). https://doi.org/10.1134/S0016266322010075
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DOI: https://doi.org/10.1134/S0016266322010075