A note on the isomorphism conjectures for Leavitt path algebras

TL;DR

This paper explores the relationship between two classification conjectures for Leavitt path algebras, showing that isomorphism of graded K-theory implies isomorphism of ungraded K-theory for certain finite graphs.

Contribution

It establishes a link between graded and ungraded K-theory classifications for Leavitt path algebras of finite graphs with no sinks.

Findings

Isomorphism of K^{ ext{gr}}_0-groups implies isomorphism of K_0-groups for certain graphs.

Supports the conjecture that K_0 classifies purely infinite simple unital Leavitt path algebras.

Provides evidence connecting two major classification conjectures.

Abstract

We relate two conjectures which have been raised for classification of Leavitt path algebras. For purely infinite simple unital Leavitt path algebras, it is conjectured that K_0 classifies them completely. For arbitrary Leavitt path algebras, it is conjectured that K^{\gr}_0 classifies them completely \cite{hazann}. We show that for two finite graphs with no sinks (which their associated Leavitt path algebras include the purely infinite simple ones) if their K^{\gr}_0-groups of their Leavitt path algebras are isomorphic then their K_0-groups are isomorphic as well.