Locally unital C^*-algebras and limitations of seminorm criteria for compact operators

Abstract

We study the problem of geometric characterization of Banach-compact and 𝒜-compact operators on Hilbert C^*-modules in terms of uniform structures generated by seminorms. Previously, such a criterion for 𝒜-compactness of adjointable operators was obtained by Troitsky for countably generated range modules with a suitable system of seminorms. Using locally unital C^*-algebras as modules over themselves, we establish fundamental limitations of such characterizations. Namely, we show that for nonunital locally unital C^*-algebras not only does Troitsky's criterion fail to extend, but, more generally, no system of seminorms can characterize 𝒜-precompactness by total boundedness, nor Banach-compactness of bounded operators or 𝒜-compactness of adjointable operators by total boundedness of the images of the unit ball. Moreover, the same statement holds for Hilbert C^*-modules 𝒩 whenever the algebra K(𝒩) of 𝒜-compact operators is locally unital and nonunital. We also study Troitsky-type systems of seminorms and their relation to 𝒜-precompactness. We prove that every 𝒜-precompact subset is totally bounded with respect to the strongest of such structures. Consequently, for Hilbert C^*-modules admitting a standard frame, all the corresponding notions of total boundedness coincide.

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BibTeX RIS

BibTeX

@misc{fufaev2026locally,
  title = {{Locally unital $C^*$-algebras and limitations of seminorm criteria for compact operators}},
  author = {Denis Fufaev},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.39164},
  eprint = {2610.39164},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.39164},
}

RIS

TY  - UNPB
TI  - Locally unital C^*-algebras and limitations of seminorm criteria for compact operators
AU  - Denis Fufaev
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2610.39164
UR  - https://arxiv.org/abs/2610.39164
AB  - We study the problem of geometric characterization of Banach-compact and 𝒜-compact operators on Hilbert C^*-modules in terms of uniform structures generated by seminorms. Previously, such a criterion for 𝒜-compactness of adjointable operators was obtained by Troitsky for countably generated range modules with a suitable system of seminorms. Using locally unital C^*-algebras as modules over themselves, we establish fundamental limitations of such characterizations. Namely, we show that for nonunital locally unital C^*-algebras not only does Troitsky's criterion fail to extend, but, more generally, no system of seminorms can characterize 𝒜-precompactness by total boundedness, nor Banach-compactness of bounded operators or 𝒜-compactness of adjointable operators by total boundedness of the images of the unit ball. Moreover, the same statement holds for Hilbert C^*-modules 𝒩 whenever the algebra K(𝒩) of 𝒜-compact operators is locally unital and nonunital. We also study Troitsky-type systems of seminorms and their relation to 𝒜-precompactness. We prove that every 𝒜-precompact subset is totally bounded with respect to the strongest of such structures. Consequently, for Hilbert C^*-modules admitting a standard frame, all the corresponding notions of total boundedness coincide.
ER  -

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