Quantum edge correspondences and quantum Cuntz-Krieger algebras

TL;DR

This paper introduces a quantum edge correspondence for quantum graphs, linking it to quantum Cuntz-Krieger algebras, and explores conditions for faithfulness and algebraic isomorphisms, with applications to examples and Exel crossed products.

Contribution

It defines a quantum edge correspondence for quantum graphs and establishes its relation to quantum Cuntz-Krieger algebras, including conditions for faithfulness and algebraic quotients.

Findings

Faithfulness of the correspondence depends on the kernel of the quantum adjacency matrix.

The Cuntz-Pimsner algebra is isomorphic to a quotient of the quantum Cuntz-Krieger algebra.

The kernel of the quotient is generated by localized quantum relations.

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 $A\colon B\to B$ 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.