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, such as Poincaré duality, which are of independent interest.