Skip to content
DraconiticResearch
Search Sources API
190,869 studies31 sourcesUpdated 3 min ago
Peer-reviewedCC BYProceedings of the London Mathematical Society

The diagonal dimension of sub‐C∗‐algebras

  • Kang Li1
  • Hung‐Chang Liao2
  • Wilhelm Winter3
  1. 1Centre for Mathematical Sciences Lund University Lund Sweden
  2. 2Department of Mathematics and Statistics University of Ottawa Ottawa Ontario Canada
  3. 3Mathematisches Institut der WWU Münster Münster Deutschland
Open at sourceDOICite

Abstract

We introduce diagonal dimension , a version of nuclear dimension for diagonal ‐algebras (sometimes also referred to as diagonal ‐pairs). Our concept has good permanence properties and detects more refined information than nuclear dimension. In many situations, it is precisely how dynamical information is encoded in an associated ‐pair. For free actions on compact Hausdorff spaces, diagonal dimension of the crossed product with its canonical diagonal is bounded above by a product involving Kerr's tower dimension of the action and covering dimension of the space. It is bounded below by the dimension of the space, by the asymptotic dimension of the group, and by the fine tower dimension of the action. For a locally compact, Hausdorff, étale groupoid, diagonal dimension of the groupoid ‐algebra is bounded below by the dynamic asymptotic dimension of the groupoid. For free Cantor dynamical systems, diagonal dimension (defined at the level of the crossed product ‐algebra) and tower dimension (an entirely dynamical notion) agree on the nose. Similarly, for a finitely generated group diagonal dimension of its uniform Roe algebra with the canonical diagonal agrees precisely with asymptotic dimension of the group. This statement also holds for uniformly bounded metric spaces. We apply the lower bounds above to a number of further examples which show how diagonal dimension keeps track of information not seen by nuclear dimension.

Related

  1. An Approximate-Diagonal Characterization of Amenable Actions on C*Algebras
    Zhengyu Fu
    arXivPreprintarXiv licenceFull textAnalysis & PDEsarXiv
  2. Isospectral Deformations of Diagonal Matrices by Symmetric Matrices in Dimension Four
    Hari Asokan
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  3. Finite-Dimensional Type I von Neumann Algebras in PyTorch: A GPU-Accelerated Framework for Random Block-Diagonal Operators
    Irina Nikolaeva, Andrej Novikov
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryAnalysis & PDEsarXiv
  4. Jacobson radical non-nil algebras of Gel'fand-Kirillov dimension 2
    Agata Smoktunowicz, Laurent Bartholdi
    Israel Journal of MathematicsPeer-reviewedLicence not statedFull text4 citedAlgebra & number theoryHAL
  5. Nuclear Dimension of Twisted C*-Algebras of Virtually Abelian Groups
    Forrest Glebe, Pradyut Karmakar, Iason Moutzouris
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryAnalysis & PDEsarXiv

Cite

BibTeXRIS
BibTeX
@article{li2026diagonal,
  title = {{The diagonal dimension of sub‐C∗‐algebras}},
  author = {Kang Li and Hung‐Chang Liao and Wilhelm Winter},
  journal = {Proceedings of the London Mathematical Society},
  year = {2026},
  month = oct,
  doi = {10.1112/plms.70234},
  url = {https://doi.org/10.1112/plms.70234},
  note = {CC BY},
}
RIS
TY  - JOUR
TI  - The diagonal dimension of sub‐C∗‐algebras
AU  - Kang Li
AU  - Hung‐Chang Liao
AU  - Wilhelm Winter
PY  - 2026
DA  - 2026/10/01
JO  - Proceedings of the London Mathematical Society
DO  - 10.1112/plms.70234
UR  - https://doi.org/10.1112/plms.70234
AB  - We introduce diagonal dimension , a version of nuclear dimension for diagonal ‐algebras (sometimes also referred to as diagonal ‐pairs). Our concept has good permanence properties and detects more refined information than nuclear dimension. In many situations, it is precisely how dynamical information is encoded in an associated ‐pair. For free actions on compact Hausdorff spaces, diagonal dimension of the crossed product with its canonical diagonal is bounded above by a product involving Kerr's tower dimension of the action and covering dimension of the space. It is bounded below by the dimension of the space, by the asymptotic dimension of the group, and by the fine tower dimension of the action. For a locally compact, Hausdorff, étale groupoid, diagonal dimension of the groupoid ‐algebra is bounded below by the dynamic asymptotic dimension of the groupoid. For free Cantor dynamical systems, diagonal dimension (defined at the level of the crossed product ‐algebra) and tower dimension (an entirely dynamical notion) agree on the nose. Similarly, for a finitely generated group diagonal dimension of its uniform Roe algebra with the canonical diagonal agrees precisely with asymptotic dimension of the group. This statement also holds for uniformly bounded metric spaces. We apply the lower bounds above to a number of further examples which show how diagonal dimension keeps track of information not seen by nuclear dimension.
SN  - 0024-6115
SN  - 1460-244X
ER  -