Pith. sign in

REVIEW 6 cited by

The diagonal dimension of sub-C*-algebras

Not yet reviewed by Pith; the record is open.

Re-run · record.json Download PDF Read on arXiv ↗

This paper has not been read by Pith yet. Machine review is queued; the pith claim, tier, and objections will appear here once it completes.

SPECIMEN: schema-true, not a live event

T0 review · schema-true

One-sentence machine reading of the paper's core claim.

pith:XXXXXXXX · record.json · timestamp

arxiv 2303.16762 v1 pith:AXJQHZJB submitted 2023-03-29 math.OA math.DS

Kang Li , Hung-Chang Liao , Wilhelm Winter This is my paper · ORCID
classification math.OAmath.DS
keywords dimensiondiagonalboundedalgebraasymptoticdynamicalgroupgroupoid
verification ladder T0 review T1 audit T2 compute T3 formal
0 comments
read the original abstract

We introduce diagonal dimension, a version of nuclear dimension for diagonal sub-C*-algebras (sometimes also referred to as diagonal C*-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 C*-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, \'etale groupoid, diagonal dimension of the groupoid C*-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 C*-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.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 6 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. A C*-diagonal in the Jiang-Su algebra via entangled matrix cones

    math.OA 2026-07 accept novelty 7.0 of 10

    An explicit inductive system of entangled dimension-drop algebras realises Z and yields a C*-diagonal with one-dimensional non-locally-connected spectrum, via a new normaliser characterisation by state excision.

  2. Functoriality and Weyl Groupoids of Ample C*-Diagonal Pairs

    math.OA 2026-05 unverdicted novelty 7.0 of 10

    The paper initiates a functorial study by constructing induced partial morphisms on Weyl groupoids from morphisms of ample C*-diagonal pairs and proves applications including tensor product identification and subaddit...

  3. Paper-folding models for the CAR algebra

    math.OA 2025-08 conditional novelty 7.0 of 10

    The CAR algebra admits countably many pairwise non-conjugate C*-diagonals with Cantor spectrum, distinguished by diagonal dimension, including non-AF examples.

  4. Dynamic asymptotic dimension growth for group actions and groupoids

    math.DS 2024-11 conditional novelty 7.0 of 10

    Dynamic asymptotic dimension growth is introduced and shown to be equivalent to asymptotic dimension growth for coarse groupoids, implying amenability for groupoids with sublinear dynamic dimension growth.

  5. Diagonal dimension and intermediate sub-C*-algebras

    math.OA 2026-07 accept novelty 6.0 of 10

    For a C*-diagonal (D⊂A) with D separable and intermediate D⊂B⊂A, dim+1_diag(D⊂B) ≤ dim+1(ˆD)·dim+1_diag(D⊂A).

  6. On a C*-Diagonal Generated by the Toric Code

    math.OA 2026-01 conditional novelty 6.0 of 10

    The toric-code stabilizer algebra is a C*-diagonal of M_{2^∞} equivalent to the canonical diagonal.

Pith tools