On the -theory of C^*-algebras associated to substitution tilings
Tom 551 / 2020
Dissertationes Mathematicae 551 (2020), 1-133
MSC: 52C23, 46L80, 37D15.
DOI: 10.4064/dm800-4-2020
Opublikowany online: 28 July 2020
Streszczenie
Under suitable conditions, a substitution tiling gives rise to a Smale space, from which three equivalence relations can be constructed, namely the stable, unstable, and asymptotic equivalence relations. We denote by S, U, and A their corresponding C^*-algebras in the sense of Renault. We show that the K-theories of S and U can be computed from the cohomology and homology of a single cochain complex with connecting maps for tilings of the line and of the plane. Moreover, we provide formulas to compute the K-theory for these three C^*-algebras. Furthermore, we show that the K-theory groups for tilings of dimension 1 are always torsion free. For tilings of dimension 2, only K_0(U) and K_1(S) can contain torsion.
