Classifying Leavitt path algebras up to involution preserving homotopy

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.

39 pages, no figures. Second version is fully re-written. Third version clarifies the relevance of classic results of Joachim Cuntz to the present work. Fourth version corrects a typo and adds reference to Cuntz' homotopy classification of Cuntz-Krieger algebras [20]