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Affine buildings, tiling systems and higher rank Cuntz-Krieger algebras. (English) Zbl 1064.46504

Summary: To an \(r\)-dimensional subshift of finite type satisfying certain special properties we associate a \(C^*\)-algebra \(\mathcal A\). This algebra is a higher-rank version of a Cuntz-Krieger algebra. In particular, it is simple, purely infinite and nuclear. We study an example: If \(\Gamma\) is a group acting freely on the vertices of an \(\tilde A_2\) building, with finitely many orbits, and if \(\Omega\) is the boundary of that building, then \(C(\Omega)\rtimes\Gamma\) is the algebra associated to a certain two-dimensional subshift.

MSC:

46L05 General theory of \(C^*\)-algebras
46L55 Noncommutative dynamical systems
20E42 Groups with a \(BN\)-pair; buildings
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