Spectral Triples on a non-standard presentation of Effros-Shen AF algebras
Konrad Aguilar, Samantha Brooker, Jack Spielberg

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 can be described by an inductive sequence of direct sums of matrix algebras, where the continued fraction expansion of encodes the dimensions of the summands, and how the matrix algebras at the th level fit into the summands at the th level. In recent work, Mitscher and Spielberg present an Effros-Shen algebra as the -algebra of a category of paths -- a generalization of a directed graph -- determined by the continued fraction expansion of . 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…
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Advanced Operator Algebra Research · Advanced Algebra and Logic
