Quantum edge correspondences and quantum Cuntz-Krieger algebras

Abstract

Given a quantum graph $\mathcal{G}=(B,\psi ,A)$, we define a C*-correspondence $E_{\mathcal{G}}$ over the noncommutative vertex C*-algebra $B$, called the quantum edge correspondence. For a classical graph $\mathcal{G}$, $E_{\mathcal{G}}$ is the usual graph correspondence spanned by the edges of $\mathcal{G}$. When the quantum adjacency matrix $A:B\to B$ is completely positive, we show that $E_{\mathcal{G}}$ is faithful if and only if $\ker (A)$ does not contain a central summand of $B$. In this case, we show that the Cuntz-Pimsner algebra ${\mathcal{O}}_{E_{\mathcal{G}}}$ is isomorphic to a quotient of the quantum Cuntz-Krieger algebra $\mathcal{O}(\mathcal{G})$ defined by Brannan, Eifler, Voigt, and Weber. Moreover, the kernel of the quotient map is shown to be generated by "localized" versions of the quantum Cuntz-Krieger relations, and ${\mathcal{O}}_{E_{\mathcal{G}}}$ is shown to be the universal object associated to these local relations. We study in detail some concrete examples and make connections with the theory of Exel crossed products.

23 pages