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

TL;DR

This work constructs a spectral triple on the Effros–Shen AF algebra $\mathcal{AF}_{\theta}$ by leveraging Mitscher–Spielberg’s category-of-paths presentation as a groupoid C*-algebra. It adapts Christensen–Ivan’s inductive-subelement strategy to an inductive limit of infinite-dimensional subalgebras, achieving a finite-bandwidth decomposition of the GNS Hilbert space and a Dirac operator that acts blockwise. Central contributions include (i) a detailed finite-dimensional block decomposition of the GNS space, (ii) a bandwidth theorem ensuring bounded commutators with the Dirac operator for a dense subalgebra, and (iii) a concrete unbounded operator $D$ yielding a valid spectral triple $(A,H,D)$ for the path-based AF algebra. The results connect noncommutative geometry with C*-algebras of categories of paths, providing a geometric framework for spectral analysis on $\mathcal{AF}_{\theta}$ with potential implications for metrics and metric data on such AF algebras.

Abstract

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.

Figures (3)

• Figure 1: $\Lambda_1$
• Figure 2: $\Lambda_2$
• Figure 3: $\Lambda$

Theorems & Definitions (74)

• Definition 2.1
• Definition 2.2
• Definition 2.3: Spi14
• Definition 2.4: Spi14
• Theorem 2.5
• Definition 2.6: Sims18
• Theorem 3.1
• Proposition 3.2
• proof
• Theorem : Theorem \ref{['thm total bandwidth']}