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) := 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 Z k ∈ L(X E k ) 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 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.