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ALGEBRAIC CUNTZ–KRIEGER ALGEBRAS

Part of: Rings and algebras with additional structure Modules, bimodules and ideals Homological methods

Published online by Cambridge University Press:  23 September 2019

ALIREZA NASR-ISFAHANI
ALIREZA NASR-ISFAHANI*
Affiliation:
Department of Mathematics, University of Isfahan, PO Box 81746-73441, Isfahan, Iran School of Mathematics, Institute for Research in Fundamental Sciences (IPM), PO Box 19395-5746, Tehran, Iran email [email protected], [email protected]
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Abstract

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.

Keywords

Leavitt path algebraCuntz–Krieger algebrastably isomorphicMorita equivalence

MSC classification

Primary: 16W99: None of the above, but in this section
Secondary: 16E20: Grothendieck groups, $K$-theory, etc. 16D90: Module categories ; module theory in a category-theoretic context; Morita equivalence and duality
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 109 , Issue 1 , August 2020 , pp. 93 - 111
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Copyright
© 2019 Australian Mathematical Publishing Association Inc.

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Footnotes

This research was in part supported by a grant from IPM (no. 95170419).

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