$C^{ast}$-Algebras Associated with Mauldin--Williams Graphs

Canadian mathematical bulletin

关键词: four-manifolds;    homotopy type;    obstruction theory;    homology with local coefficients;    surgery;    normal invariant;    assembly map;   

DOI  :  10.4153/CMB-2008-054-0

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

A Mauldin--Williams graph $mathcal{M}$ is a generalization of aniterated function system by a directed graph. Its invariant set $K$plays the role of the self-similar set. We associate a $C^{*}$-algebra$mathcal{O}_{mathcal{M}}(K)$ with a Mauldin--Williams graph $mathcal{M}$and the invariant set $K$, laying emphasis on the singular points.We assume that the underlying graph $G$ has no sinks and no sources.If $mathcal{M}$ satisfies the open set condition in $K$, and $G$is irreducible and is not a cyclic permutation, then the associated$C^{*}$-algebra $mathcal{O}_{mathcal{M}}(K)$ is simple and purelyinfinite. We calculate the $K$-groups for some examples including theinflation rule of the Penrose tilings.

【 授权许可】

Unknown