Spectral Triples on a non-standard presentation of Effros-Shen AF algebras

TL;DR

This paper constructs spectral triples for Effros-Shen AF algebras using a path category approach, extending previous finite-dimensional models to an infinite-dimensional setting based on continued fraction expansions.

Contribution

It introduces a spectral triple framework for Effros-Shen algebras via a path category representation, generalizing earlier finite-dimensional constructions.

Findings

Spectral triples are successfully constructed for Effros-Shen algebras.

The approach extends finite-dimensional models to infinite-dimensional cases.

Provides a new perspective on the structure of Effros-Shen algebras.

Abstract

The Effros-Shen algebra corresponding to an irrational number $\theta$ can be described by an inductive sequence of direct sums of matrix algebras, where the continued fraction expansion of $\theta$ 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 $\theta$. 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.