Filtered K-theory for graph algebras

TL;DR

This paper develops a filtered algebraic K-theory framework for rings, especially Leavitt path algebras, paralleling graph C*-algebra theory, and uses it to verify a conjecture for many finite graphs.

Contribution

It introduces a new filtered algebraic K-theory for rings relative to ideals, aligning with graph C*-algebra theory, and applies it to confirm the Abrams-Tomforde conjecture.

Findings

Filtered algebraic K-theory parallels gauge invariant K-theory for graph C*-algebras.

Verified the Abrams-Tomforde conjecture for a large class of finite graphs.

Established a new method for analyzing Leavitt path algebras using filtered K-theory.

Abstract

We introduce filtered algebraic $K$-theory of a ring $R$ relative to a sublattice of ideals. This is done in such a way that filtered algebraic $K$-theory of a Leavitt path algebra relative to the graded ideals is parallel to the gauge invariant filtered $K$-theory for graph $C^*$-algebras. We apply this to verify the Abrams-Tomforde conjecture for a large class of finite graphs.