Quantum edge correspondences and quantum Cuntz–Krieger algebras

Given a quantum graph G=(B,ψ,A)${\mathcal {G}}=(B,\psi ,A)$, we define a C*-correspondence EG$E_{\mathcal {G}}$ over the non-commutative vertex C*-algebra B$B$, called the quantum edge correspondence. For a classical graph G${\mathcal {G}}$, EG$E_{\mathcal {G}}$ is the usual graph correspondence spanned by the edges of G${\mathcal {G}}$. When the quantum adjacency matrix A:B→B$A\colon B\rightarrow B$ is completely positive, we show that EG$E_{\mathcal {G}}$ is faithful if and only if ker⁡(A)$\ker (A)$ does not contain a central summand of B$B$. In this case, we show that the Cuntz–Pimsner algebra OEG${\mathcal {O}}_{E_{\mathcal {G}}}$ is isomorphic to a quotient of the quantum Cuntz–Krieger algebra O⁡(G)${\mathcal {O}}({\mathcal {G}})$ defined in Brannan, Eifler, Voigt, and Weber (Trans. Am. Math. Soc. Ser. B 9 (2022), 782–826). Moreover, the kernel of the quotient map is shown to be generated by “localized” versions of the quantum Cuntz–Krieger relations, and OEG${\mathcal {O}}_{E_{\mathcal {G}}}$ 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.

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