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ALGEBRAIC CUNTZ–KRIEGER ALGEBRAS
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 109 / Issue 1 / August 2020
- Published online by Cambridge University Press:
- 23 September 2019, pp. 93-111
- Print publication:
- August 2020
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- Article
- Export citation
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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.
