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

TL;DR

This paper establishes a correspondence between $C^*$-algebra homomorphisms and Leavitt path algebra homomorphisms, showing that algebraic maps can be lifted to $C^*$-algebra maps and analyzing their homotopy properties.

Contribution

It proves that unital $*$-homomorphisms between certain $C^*$-algebras can be lifted from algebraic Leavitt path algebra maps, linking algebraic and topological homotopy equivalences.

Findings

Existence of algebraic homomorphisms lifting $C^*$-algebra maps.

Homotopy equivalence in $C^*$-algebras corresponds to algebraic homotopy.

Any isomorphism between simple purely infinite Cuntz-Krieger algebras is homotopic to an algebraic homotopy.

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.