\(C^*\)-algebras associated with presentations of subshifts
Let \(\Sigma\) be a finite set. A \(\lambda\)-graph system \(L=(V,E,\lambda,\iota)\) consists of a vertex set \(V=V_0\cup V_1\cup V_2\cup\ldots\), an edge set \(E=E_{0,1}\cup E_{1,2}\cup E_{2,3}\cup\ldots\), a labeling map \(\lambda:E\to\Sigma\) and a surjection \(\iota_i:V_i\to V_{i-1}\) for each \(i\in\mathbb N\). The author introduces \(C^*\)-algebras from \(\lambda\)-graph systems. If a \(\lambda\)-graph system is the canonical \(\lambda\)-graph system for a subshift \(\Lambda\), then the corresponding \(C^*\)-algebra coincides with the Cuntz-Krieger algebra \({\mathcal O}_\Lambda\) associated with \(\Lambda\). If a \(\lambda\)-graph system is irreducible (resp. aperiodic), its \(C^*\)-algebra is simple (resp. simple purely infinite). An example of a simple purely infinite \(C^*\)-algebra of a \(\lambda\)-graph system that is not stably isomorphic to any Cuntz-Krieger algebra is produced.
scientific article; zbMATH DE number 1748373