Spectral Triples on a non-standard presentation of Effros-Shen AF algebras
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912662562013184 |
|---|---|
| author | Aguilar, Konrad Brooker, Samantha Spielberg, Jack |
| author_facet | Aguilar, Konrad Brooker, Samantha Spielberg, Jack |
| contents | The Effros-Shen algebra corresponding to an irrational number $θ$ can be described by an inductive sequence of direct sums of matrix algebras, where the continued fraction expansion of $θ$ encodes the dimensions of the summands, and how the matrix algebras at the $n$th level fit into the summands at the $(n+1)$th level. In recent work, Mitscher and Spielberg present an Effros-Shen algebra as the $C^*$-algebra of a category of paths -- a generalization of a directed graph -- determined by the continued fraction expansion of $θ$. With this approach, the algebra is realized as the inductive limit of a sequence of infinite-dimensional, rather than finite-dimensional, subalgebras. In the present work, we define a spectral triple in terms of the category of paths presentation of an Effros-Shen algebra, drawing on a construction by Christensen and Ivan. This article describes categories of paths, the example of Mitscher and Spielberg, and the spectral triple construction. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_18146 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spectral Triples on a non-standard presentation of Effros-Shen AF algebras Aguilar, Konrad Brooker, Samantha Spielberg, Jack Operator Algebras Functional Analysis 46L05 (Primary), 46L87, 58B34 (Secondary) The Effros-Shen algebra corresponding to an irrational number $θ$ can be described by an inductive sequence of direct sums of matrix algebras, where the continued fraction expansion of $θ$ encodes the dimensions of the summands, and how the matrix algebras at the $n$th level fit into the summands at the $(n+1)$th level. In recent work, Mitscher and Spielberg present an Effros-Shen algebra as the $C^*$-algebra of a category of paths -- a generalization of a directed graph -- determined by the continued fraction expansion of $θ$. With this approach, the algebra is realized as the inductive limit of a sequence of infinite-dimensional, rather than finite-dimensional, subalgebras. In the present work, we define a spectral triple in terms of the category of paths presentation of an Effros-Shen algebra, drawing on a construction by Christensen and Ivan. This article describes categories of paths, the example of Mitscher and Spielberg, and the spectral triple construction. |
| title | Spectral Triples on a non-standard presentation of Effros-Shen AF algebras |
| topic | Operator Algebras Functional Analysis 46L05 (Primary), 46L87, 58B34 (Secondary) |
| url | https://arxiv.org/abs/2510.18146 |