Lifting graph $C^*$-algebra maps to Leavitt path algebra maps

Abstract: Let $\xi:C^*(E)\to C^*(F)$ be a unital $$-homomorphism between simple purely infinite Cuntz-Krieger algebras of finite graphs. We prove that there exists a unital $$-homomorphism $\phi:L(E)\to L(F)$ between the corresponding Leavitt path-algebras such that $\xi$ is homotopic to the map $\hat{\phi}:C^*(E)\to C^*(F)$ induced by completion. We show moreover that $\hat{\phi}$ is a homotopy equivalence in the $C^*$-algebraic sense if and only if $\phi$ 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.