论文标题
量子边缘对应关系和量子Cuntz-Krieger代数
Quantum edge correspondences and quantum Cuntz-Krieger algebras
论文作者
论文摘要
给定一个量子图$ \ MATHCAL {g} =（b，ψ，a）$，我们在非交换Vertex c*-algebra $ b $上定义了a c*-correspess $ e_ \ nathcal {g} $，称为量子边缘通讯。对于经典图形$ \ MATHCAL {g} $，$ e_ \ Mathcal {g} $是$ \ Mathcal {G} $的边缘所跨越的通常图形通信。当量子邻接矩阵$ a \ colon b \ to b $完全阳性时，我们表明$ e_ \ mathcal {g} $在且仅当$ \ ker（a）$中不包含$ b $的中心总汇总时。 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.此外，商映射的内核被证明是由量子cuntz-krieger关系的“局部”版本生成的，$ \ Mathcal {o} _ {e_ \ Mathcal {g}} $显示为与这些本地关系相关的通用对象。我们详细研究一些具体的例子，并与Exel交叉产品理论建立联系。
Given a quantum graph $\mathcal{G}=(B,ψ,A)$, 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.
