The homotopy lifting theorem for semiprojective C*-algebras

TL;DR

This paper establishes a homotopy lifting theorem for semiprojective C*-algebras, extending classical topological results to the operator algebra setting and providing new lifting and homotopy results.

Contribution

It introduces a homotopy lifting theorem for semiprojective C*-algebras and related results on homomorphism lifting and homotopy, advancing the theory of operator algebras.

Findings

Proved a homotopy lifting theorem for semiprojective C*-algebras.

Established a partial lifting theorem with specified quotient.

Showed that sufficiently close homomorphisms are homotopic.

Abstract

We prove a complete analog of the Borsuk Homotopy Extension Theorem for arbitrary semiprojective C*-algebras. We also obtain some other results about semiprojective C*-algebras: a partial lifting theorem with specified quotient, a lifting result for homomorphisms close to a liftable homomorphism, and that sufficiently close homomorphisms from a semiprojective C*-algebra are homotopic.