Classifying Leavitt path algebras up to involution preserving homotopy

TL;DR

This paper demonstrates that the Bowen-Franks group uniquely classifies purely infinite simple Leavitt path algebras over certain rings up to involution-preserving homotopy, using $K$-theoretic methods.

Contribution

It establishes classification results for Leavitt path algebras via $K$-theory and explores involution-preserving homotopy equivalences, including twisted involutions.

Findings

Bowen-Franks group classifies Leavitt path algebras over specified rings.

Results hold for purely infinite simple finite graphs with standard involutions.

Partial classification results for twisted involutions on these algebras.

Abstract

We prove that the Bowen-Franks group classifies the Leavitt path algebras of purely infinite simple finite graphs over a regular supercoherent commutative ring with involution where $2$ is invertible, equipped with their standard involutions, up to matricial stabilization and involution preserving homotopy equivalence. We also consider a twisting of the standard involution on Leavitt path algebras and obtain partial results in the same direction for purely infinite simple graphs. Our tools are $K$-theoretic, and we prove several results about (Hermitian, bivariant) $K$-theory of Leavitt path algebras.