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

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

2008-04-22

arxiv.org/abs/0804.3447v1

Mathematics

Operator Algebras

28 pages, 3 figures, 1 table

Scientific paper

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.

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