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 $π\colon E \rightarrow B$ and for every $^*$-homomorphism $φ\colon A \rightarrow E$, any path of $^*$-homomorphisms $A \rightarrow B$ starting at $πφ$ lifts to a path of $^*$-homomorphisms $A \rightarrow E$ starting at $φ$. 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 $π$ satisfies an approximate decomposition property in the spirit of (but weaker than) the notion of quasidiagonality for extensions.

Theorems & Definitions (51)

• Theorem A
• Theorem B: cf. Corollary \ref{['cor:ahlp-qdl']}
• Definition 1
• Proposition 2
• proof
• Definition 3
• Definition 4
• Lemma 5
• proof
• Lemma 6