Quantum edge correspondences and quantum Cuntz–Krieger algebras
Quantum edge correspondences and quantum Cuntz–Krieger algebras
复制标题
量子边对应和量子 Cuntz Krieger 代数
DOI：
10.1112/jlms.12702
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发表时间：
2023
期刊：
影响因子：
--
通讯作者：
Wasilewski, Mateusz
中科院分区：
文献类型：
--
作者：
Brannan, Michael;Hamidi, Mitch;Ismert, Lara;Nelson, Brent;Wasilewski, Mateusz
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 thequantum 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. B9(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.
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 thequantum 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. B9(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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DOI：
10.1109/focs46700.2020.00067
发表时间：
2019-10
期刊：
2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS)
影响因子：
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作者：
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DOI：
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期刊：
Canadian Journal of Mathematics
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影响因子：
1.1
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