The Algebraic Kirchberg-Phillips Question for Leavitt path algebras

TL;DR

This paper investigates whether unital $K$-theory fully classifies unital, simple, purely infinite Leavitt path algebras over finite graphs, linking specific non-simple cases to the broader classification problem.

Contribution

It introduces a new question about isomorphism of particular non-simple Leavitt path algebras and proves its positive answer implies a positive solution to the Algebraic Kirchberg-Phillips Question.

Findings

Positively answering the new isomorphism question implies the algebraic classification conjecture.

Links specific non-simple algebra isomorphisms to the broader classification problem.

Provides a new approach to the algebraic Kirchberg-Phillips Question.

Abstract

The Algebraic Kirchberg-Phillips Question for Leavitt path algebras asks whether unital $K$-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 $L_k(\mathbf{F}_*)$ and $L_k(\mathbf{F}_{**})$ are isomorphic, and we prove that a positive answer to this question implies a positive answer to the Algebraic Kirchberg-Phillips Question.