## 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 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.

Original language | English |
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Article number | 37 |

Journal | Integral Equations and Operator Theory |

Volume | 94 |

Issue number | 4 |

DOIs | |

State | Published - 1 Dec 2022 |

## Keywords

- C-algebra
- C-correspondence
- Cuntz–Krieger algebras
- Cuntz–Pimsner algebra
- Directed graph
- Fock space
- Gauge-invariant ideals
- Graph algebras
- Simplicity
- Weighted shift

## ASJC Scopus subject areas

- Analysis
- Algebra and Number Theory