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

Abstract

Higher-rank graphs (or $k$-graphs) were introduced by Kumjian and Pask to provide combinatorial models for the higher-rank Cuntz-Krieger $C^{\ast }$-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 $C^{\ast }$-algebras of these 2-graphs, find criteria under which they are simple and purely infinite, and compute their $K$-theory. We find examples whose $C^{\ast }$-algebras satisfy the hypotheses of the classification theorem of Kirchberg and Phillips, but are not isomorphic to the $C^{\ast }$-algebras of ordinary directed graphs.

28 pages, 3 figures, 1 table

References in corpus (1)

• Simplicity of C*-algebras associated to higher-rank graphs