Weighted Cuntz–Krieger Algebras

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) : = T(X_E, Z) / K where T(X_E, Z) is the C^∗-algebra generated by weighted shifts on the Fock correspondence F(X_E) given by a weight sequence { Z_k} of operators Zk∈L(XEk) and 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 (the vertices of E) to be hereditary or saturated. As an example, we discuss in some details the case where E is a cycle.