We show that a directed graph $E$ is a finite graph with no sinks if and only if, for each commutative unital ring $R$, the Leavitt path algebra $L_{R}(E)$ is isomorphic to an algebraic Cuntz–Krieger algebra if and only if the $C^{\ast }$-algebra $C^{\ast }(E)$ is unital and $\text{rank}(K_{0}(C^{\ast }(E)))=\text{rank}(K_{1}(C^{\ast }(E)))$. Let $k$ be a field and $k^{\times }$ be the group of units of $k$. When $\text{rank}(k^{\times })<\infty$, we show that the Leavitt path algebra $L_{k}(E)$ is isomorphic to an algebraic Cuntz–Krieger algebra if and only if $L_{k}(E)$ is unital and $\text{rank}(K_{1}(L_{k}(E)))=(\text{rank}(k^{\times })+1)\text{rank}(K_{0}(L_{k}(E)))$. We also show that any unital $k$-algebra which is Morita equivalent or stably isomorphic to an algebraic Cuntz–Krieger algebra, is isomorphic to an algebraic Cuntz–Krieger algebra. As a consequence, corners of algebraic Cuntz–Krieger algebras are algebraic Cuntz–Krieger algebras.

中文翻译：

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代数康茨-克里格代数

我们证明了一个有向图$E$是一个没有汇的有限图当且仅当，对于每个交换单位环$R$, Leavitt 路径代数$L_{R}(E)$与代数 Cuntz-Krieger 代数同构当且仅当$C^{\ast }$-代数$C^{\ast }(E)$是统一的并且$\text{rank}(K_{0}(C^{\ast }(E)))=\text{rank}(K_{1}(C^{\ast }(E)))$. 让$k$成为一个领域并且$k^{\times }$是单位的群$k$. 什么时候$\text{rank}(k^{\times })<\infty$, 我们证明了 Leavitt 路径代数$L_{k}(E)$与代数 Cuntz-Krieger 代数同构当且仅当$L_{k}(E)$是统一的并且$\text{rank}(K_{1}(L_{k}(E)))=(\text{rank}(k^{\times })+1)\text{rank}(K_{0}( L_{k}(E)))$. 我们还表明，任何单位$k$-代数与代数 Cuntz-Krieger 代数等价或稳定同构，与代数 Cuntz-Krieger 代数同构。因此，代数 Cuntz-Krieger 代数的角是代数 Cuntz-Krieger 代数。