Weighted Cuntz–Krieger Algebras

Abstract

Let E be a finite directed graph with no sources or sinks and write \(X_E\) for the graph correspondence. We study the \(C^*\)-algebra \(C^*(E,Z):=\mathcal {T}(X_E,Z)/\mathcal {K}\) where \(\mathcal {T}(X_E,Z)\) is the \(C^*\)-algebra generated by weighted shifts on the Fock correspondence \(\mathcal {F}(X_E)\) given by a weight sequence \(\{Z_k\}\) of operators \(Z_k\in \mathcal {L}(X_{E^{k}})\) and \(\mathcal {K}\) is the algebra of compact operators on the Fock correspondence. If \(Z_k=I\) for every k, \(C^*(E,Z)\) is the Cuntz–Krieger algebra associated with the graph E. We show that \(C^*(E,Z)\) can be realized as a Cuntz–Pimsner algebra and use a result of Schweizer to find conditions for the algebra \(C^*(E,Z)\) to be simple. We also analyse the gauge-invariant ideals of \(C^*(E,Z)\) using a result of Katsura and conditions that generalize the conditions of subsets of \(E^0\) (the vertices of E) to be hereditary or saturated. As an example, we discuss in some details the case where E is a cycle.