An asymptotic homotopy lifting property

Abstract

A $C^*$-algebra $A$ is said to have the homotopy lifting property if for all $C^*$-algebras $B$ and $E$, for every surjective $^*$-homomorphism $\pi \colon E \rightarrow B$ and for every $^*$-homomorphism $\phi \colon A \rightarrow E$, any path of $^*$-homomorphisms $A \rightarrow B$ starting at $\pi \phi$ lifts to a path of $^*$-homomorphisms $A \rightarrow E$ starting at $\phi$. Blackadar has shown that this property holds for all semiprojective $C^*$-algebras. We show that a version of the homotopy lifting property for asymptotic morphisms holds for separable $C^*$-algebras that are sequential inductive limits of semiprojective $C^*$-algebras. It also holds for any separable $C^*$-algebra if the quotient map $\pi$ satisfies an approximate decomposition property in the spirit of (but weaker than) the notion of quasidiagonality for extensions.