2021/08/12 by Helmer, Leonid, Solel, Baruch
#46L05 #46L08 #46L35 #46L89 #47L80 #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.2108.05601
Let E be a finite directed graph with no sources or sinks and write XE for the graph correspondence. We study the C^*-algebra C^*(E,Z):=T(XE,Z)/K where T(XE,Z) is the C^*-algebra generated by weighted shifts on the Fock correspondence F(XE) given by a weight sequence \Zk\ of operators Zk∈ L(XEk) and K is the algebra of compact operators on the Fock correspondence. If Zk=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 E0 (the vertices of E) to be hereditary or saturated. As an example, we discuss in some details the case where E is a cycle.