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 $C^*(E,Z):=\mathcal{T}(X_E,Z)/\mathcal{K}$ 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.