Which compacta are noncommutative ARs?

TL;DR

This paper characterizes when the $C^*$-algebra of continuous functions on a compact space is projective in a certain category, showing it occurs precisely for dendrites, which are one-dimensional absolute retracts.

Contribution

It provides a complete characterization of noncommutative ARs by linking projectivity of $C(X)$ to the topological structure of $X$ as a dendrite.

Findings

$C(X)$ is projective iff $X$ is a dendrite.

Dendrites are exactly the compacta with $ ext{dim} ext{X} extless= 1$ that are absolute retracts.

The result bridges topological properties of $X$ with algebraic properties of $C(X)$.

Abstract

We give a short answer to the question in the title: {\em dendrits}. Precisely we show that the $C^{\ast}$-algebra $C(X)$ of all complex-valued continuous functions on a compactum $X$ is projective in the category ${\mathcal C}^{1}$ of all (not necessarily commutative) unital $C^{\ast}$-algebras if and only if $X$ is an absolute retract of dimension $\dim X \leq 1$ or, equivalently, that $X$ is a dendrit.