Crossed products of the Cantor set by free minimal actions of \(\mathbb Z^d\). (English) Zbl 1084.46056
Summary: Let \(d\) be a positive integer, let \(X\) be the Cantor set, and let \(\mathbb Z^d\) act freely and minimally on \(X\). We prove that the crossed product \(C*(\mathbb Z^d,X)\) has stable rank one, real rank zero, and cancellation of projections, and that the order on \(K_0(C*(\mathbb Z^d,X))\) is determined by traces. We obtain the same conclusion for the \(C*\)-algebras of various kinds of aperiodic tilings.
MSC:
| 46L55 | Noncommutative dynamical systems |
| 46L80 | \(K\)-theory and operator algebras (including cyclic theory) |
| 37A55 | Dynamical systems and the theory of \(C^*\)-algebras |
| 37B50 | Multi-dimensional shifts of finite type, tiling dynamics (MSC2010) |
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