Weighted Cuntz–Krieger Algebras

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