2026-05-24T06:10:31Zhttps://keep.lib.asu.edu/oai/requestoai:keep.lib.asu.edu:node-1936202024-12-23T18:01:48Zoai_pmh:alloai_pmh:repo_items193620
https://hdl.handle.net/2286/R.2.N.193620
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2024
64 pages
Doctoral Dissertation
Academic theses
Text
eng
Brooker, Samantha
Spielberg, Jack
Aguilar, Konrad
Quigg, John
Kaliszewski, Steven
Paupert, Julien
Arizona State University
Partial requirement for: Ph.D., Arizona State University, 2024
Field of study: Mathematics
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 nth 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 this thesis, the author defines 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 thesis describes categories of paths, the example of Mitscher and Spielberg, and the spectral triple construction.
Theoretical mathematics
C*-algebras
operator algebras
quantum metrics
spectral triples
Spectral Triples on a Non-standard Presentation of Effros-Shen AF Algebras