The Algebraic Kirchberg–Phillips Question for Leavitt path algebras

Abstract

The Algebraic Kirchberg–Phillips Question for Leavitt path algebras asks whether pointed $K$-theory is a complete isomorphism invariant for unital, simple, purely infinite Leavitt path algebras over finite graphs. Most work on this problem has focused on determining whether (up to isomorphism) there is a unique unital, simple, Leavitt path algebra with trivial $K$-theory (often reformulated as the question of whether the Leavitt path algebras $L_2$ and $L_{2_{-}}$ are isomorphic). However, it is unknown whether a positive answer to this special case implies a positive answer to the Algebraic Kirchberg–Phillips Question. In this note, we pose a different question that asks whether two particular non-simple Leavitt path algebras $L_k\left(F_{\ast }\right)$ and $L_k\left(F_{\ast \ast }\right)$ are isomorphic, and we prove that a positive answer to this question implies a positive answer to the Algebraic Kirchberg–Phillips Question.