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.

Récupéré en direct depuis OpenAlex et désinversé. Les résumés ne sont pas conservés dans cette base de données : les index inversés représentent 8,6 Go des 9,3 Go de texte de la base, et le serveur dispose de 13 Go libres.