Lifting graph C∗-algebra maps to Leavitt path algebra maps

Abstract

Let (Formula presented.) be a unital (Formula presented.) -homomorphism between simple purely infinite Cuntz–Krieger algebras of finite graphs. We prove that there exists a unital (Formula presented.) -homomorphism (Formula presented.) between the corresponding Leavitt path-algebras such that (Formula presented.) is homotopic to the map (Formula presented.) induced by completion. We show moreover that (Formula presented.) is a homotopy equivalence in the (Formula presented.) -algebraic sense if and only if (Formula presented.) is a homotopy equivalence in the algebraic, polynomial sense. We deduce, in particular, that any isomorphism between simple purely infinite Cuntz–Krieger algebras is homotopic to the completion of a unital algebraic homotopy equivalence.