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PreprintCC BY 4.0arXiv

Certain Cuntz semigroup properties of extension C*-algebras

  • Qingzhai Fan
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Abstract

Let 0 xrightarrowI xrightarrow A xrightarrow A/I xrightarrow 0 be a short exact sequence of C*-algebras. In this paper, we prove that if dim(Cu(I))≤ n and dim(Cu(A/I))≤ m, then dim(Cu(A))≤ n+m+1, this result gives an affirmative answer to a problem concerning the Cu-semigroup of C*-algebras raised by Thiel and Vilalta. We further show that certain comparison and divisibility properties satisfied by I and A/I are inherited by the extension algebra A. As an application, we provide an alternative proof of equivalence that A is pure if and only if I and A/I, a result originally proven by Perera, Thiel, and Vilalta.

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Cite

BibTeXRIS
BibTeX
@misc{fan2026certain,
  title = {{Certain Cuntz semigroup properties of extension C*-algebras}},
  author = {Qingzhai Fan},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.27694},
  eprint = {2609.27694},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.27694},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Certain Cuntz semigroup properties of extension C*-algebras
AU  - Qingzhai Fan
PY  - 2026
DA  - 2026/09/23
PB  - arXiv
DO  - 10.48550/arxiv.2609.27694
UR  - https://arxiv.org/abs/2609.27694
AB  - Let 0 xrightarrowI xrightarrow A xrightarrow A/I xrightarrow 0 be a short exact sequence of C*-algebras. In this paper, we prove that if dim(Cu(I))≤ n and dim(Cu(A/I))≤ m, then dim(Cu(A))≤ n+m+1, this result gives an affirmative answer to a problem concerning the Cu-semigroup of C*-algebras raised by Thiel and Vilalta. We further show that certain comparison and divisibility properties satisfied by I and A/I are inherited by the extension algebra A. As an application, we provide an alternative proof of equivalence that A is pure if and only if I and A/I, a result originally proven by Perera, Thiel, and Vilalta.
ER  -