Continued fractions, directed graphs, and defining spectral triples on Effros-Shen AF algebras (Samantha Brooker, Arizona State University)

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 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 \theta. With this approach, the algebra is realized as the inductive limit of a sequence of infinite-dimensional, rather than finite-dimensional, subalgebras. Drawing on a construction by Christensen and Ivan, we use this inductive limit structure to define a spectral triple, trading the advantages of working with finite-dimensional approximants for the techniques provided by the category of paths, pursuant to studying the algebras as quantum compact metric spaces. I will discuss categories of paths and their precursors, graph C*-algebras, the example of Mitscher and Spielberg, and a bit about the spectral triple construction. This is joint work with Konrad Aguilar and Jack Spielberg.