Simplicity of Cuntz-Pimsner algebras of quantum graphs

Abstract

Let $\mathcal{G}$ be a quantum graph without quantum sources and $E_{\mathcal{G}}$ be the quantum edge correspondence for $\mathcal{G}.$ Our main results include sufficient conditions for simplicity of the Cuntz-Pimsner algebra ${\mathcal{O}}_{E_{\mathcal{G}}}$ in terms of $\mathcal{G}$ and for defining a surjection from the quantum Cuntz-Krieger algebra $\mathcal{O}(\mathcal{G})$ onto a particular relative Cuntz-Pimsner algebra for $E_{\mathcal{G}}$. 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 ${\mathcal{O}}_{E_{\mathcal{G}}}$ for some fundamental examples of quantum graphs, including rank-one 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 $E_{\mathcal{G}}$ and sufficient conditions for aperiodicity of $E_{\mathcal{G}}$ in terms of the underlying quantum graph $\mathcal{G}$.

18 pages