The homotopy lifting theorem for semiprojective C*-algebras

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.

💡 Research Summary

The paper establishes a full non‑commutative analogue of Borsuk’s Homotopy Extension Theorem for arbitrary semiprojective C*‑algebras. The author’s main result, the Homotopy Lifting Theorem, states that if (A) is semiprojective, (B) is any C*‑algebra, (I\subset B) a closed ideal, and (\phi_{0},\phi_{1}:A\to B/I) are two *‑homomorphisms together with a homotopy (H:A\to C(