Lifting graph $C^{\ast }$-algebra maps to Leavitt path algebra maps

Abstract

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

11 pages

References in corpus (2)

• K-theoretic duality for shifts of finite type
• $K$-theory of Leavitt path algebras