Algebraic Cuntz-Krieger algebras

TL;DR

This paper characterizes algebraic Cuntz-Krieger algebras via graph properties and K-theory, establishing isomorphisms with Leavitt path algebras and $C^*$-algebras, and explores their structural stability.

Contribution

It provides new criteria for identifying algebraic Cuntz-Krieger algebras using graph conditions and K-theoretic invariants, including Morita equivalence and corner stability.

Findings

Finite graphs with no sinks correspond to algebraic Cuntz-Krieger algebras.

Isomorphism conditions involve unitality and K-theory rank equalities.

Corners of these algebras are also algebraic Cuntz-Krieger algebras.

Abstract

We show that $E$ is a finite graph with no sinks if and only if the Leavitt path algebra $L_R(E)$ is isomorphic to an algebraic Cuntz-Krieger algebra if and only if the $C^*$-algebra $C^*(E)$ is unital and $rank(K_0(C^*(E)))=rank(K_1(C^*(E)))$. When $k$ is a field and $rank(k^{\times})< \infty$, 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 $rank(K_1(L_k(E)))=(rank(k^{\times})+1)rank(K_0(L_k(E)))$. 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.