Simplicity of Cuntz–Pimsner algebras of quantum graphs

Abstract Let be a quantum graph without quantum sources and be the quantum edge correspondence for . Our main results include sufficient conditions for simplicity of the Cuntz–Pimsner algebra in terms of and for defining a surjection from the quantum Cuntz–Krieger algebra onto a particular relative Cuntz–Pimsner algebra for . As an application of these two results, we give the first example of a quantum graph with distinct quantum Cuntz–Krieger and local quantum Cuntz–Krieger algebras. We also characterize simplicity of for some fundamental examples of quantum graphs, including quantum graphs on a single full matrix algebra, complete quantum graphs, and trivial quantum graphs. Along the way, we provide an equivalent condition for minimality of and sufficient conditions for aperiodicity of in terms of the underlying quantum graph .