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PreprintCC BY 4.0arXiv

Cartan Subalgebras of Dimension Drop Interval Algebras and Their Inductive Limits

  • Xinlong Cui
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Abstract

We describe the conjugacy of non-degenerate Cartan subalgebras of the dimension drop interval algebras Mm(𝕀̃p) in terms of double cosets of permutation groups, and describe their spectra by finite bi-partite multigraphs. All Cartan subalgebras are non-degenerate precisely when the two parameters are coprime. We also give splitting interval examples with prescribed finite numbers of Cartan conjugacy classes. Passing to inductive limits, we construct a unital simple AD algebra π’œ of real rank zero and stable rank one, with K0(π’œ)β‰…β„€[1/3] and K1(π’œ)β‰…β„€/2β„€. It contains infinitely many mutually isomorphic but pairwise nonconjugate C*-diagonals. Their spectra are all homeomorphic to the dense amalgam of circles, while orbit multiplicities in the Weyl groupoids distinguish their embeddings. The same AD algebra also contains infinitely many pairwise nonisomorphic Cartan subalgebras.

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Cite

BibTeXRIS
BibTeX
@misc{cui2026cartan,
  title = {{Cartan Subalgebras of Dimension Drop Interval Algebras and Their Inductive Limits}},
  author = {Xinlong Cui},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.29539},
  eprint = {2610.29539},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.29539},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Cartan Subalgebras of Dimension Drop Interval Algebras and Their Inductive Limits
AU  - Xinlong Cui
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2610.29539
UR  - https://arxiv.org/abs/2610.29539
AB  - We describe the conjugacy of non-degenerate Cartan subalgebras of the dimension drop interval algebras M_m(𝕀̃_p) in terms of double cosets of permutation groups, and describe their spectra by finite bi-partite multigraphs. All Cartan subalgebras are non-degenerate precisely when the two parameters are coprime. We also give splitting interval examples with prescribed finite numbers of Cartan conjugacy classes. Passing to inductive limits, we construct a unital simple AD algebra π’œ of real rank zero and stable rank one, with K_0(π’œ)β‰…β„€[1/3] and K_1(π’œ)β‰…β„€/2β„€. It contains infinitely many mutually isomorphic but pairwise nonconjugate C^*-diagonals. Their spectra are all homeomorphic to the dense amalgam of circles, while orbit multiplicities in the Weyl groupoids distinguish their embeddings. The same AD algebra also contains infinitely many pairwise nonisomorphic Cartan subalgebras.
ER  -