A family of 2-graphs arising from two-dimensional subshifts

TL;DR

This paper studies a specific family of 2-graphs derived from algebraic dynamical systems, analyzing their associated $C^*$-algebras for simplicity, pure infiniteness, and $K$-theory, revealing new distinctions from traditional graph $C^*$-algebras.

Contribution

It introduces a new family of 2-graphs from algebraic dynamical systems and characterizes their $C^*$-algebras, including criteria for simplicity and pure infiniteness, and computes their $K$-theory.

Findings

Identified conditions for simplicity and pure infiniteness of the $C^*$-algebras.

Computed $K$-theory for these 2-graph $C^*$-algebras.

Found examples not isomorphic to ordinary graph $C^*$-algebras despite classification hypotheses.

Abstract

Higher-rank graphs (or $k$-graphs) were introduced by Kumjian and Pask to provide combinatorial models for the higher-rank Cuntz-Krieger $C^*$-algebras of Robertson and Steger. Here we consider a family of finite 2-graphs whose path spaces are dynamical systems of algebraic origin, as studied by Schmidt and others. We analyse the $C^*$-algebras of these 2-graphs, find criteria under which they are simple and purely infinite, and compute their $K$-theory. We find examples whose $C^*$-algebras satisfy the hypotheses of the classification theorem of Kirchberg and Phillips, but are not isomorphic to the $C^*$-algebras of ordinary directed graphs.