A characterization of semiprojectivity for commutative C*-algebras

TL;DR

This paper characterizes when commutative C*-algebras are semiprojective based on the topological properties of their underlying spaces, confirming conjectures and extending results in the field.

Contribution

It provides a complete topological characterization of semiprojectivity for commutative C*-algebras, confirming Blackadar's conjecture and generalizing to the non-unital case.

Findings

C(X) is semiprojective iff X is an absolute neighborhood retract of dimension ≤ 1

Confirmed conjectures of Loring and Blackadar in the commutative case

Provided partial answers to when commutative C*-algebras are weakly (semi-)projective

Abstract

Given a compact, metric space X, we show that the commutative C*-algebra C(X) is semiprojective if and only if X is an absolute neighborhood retract of dimension at most one. This confirms a conjecture of Blackadar. Generalizing to the non-unital setting, we derive a characterization of semiprojectivity for separable, commutative C*-algebras. As further application of our findings we verify two conjectures of Loring and Blackadar in the commutative case, and we give a partial answer to the question, when a commutative C*-algebra is weakly (semi-)projective.