Which compacta are noncommutative ARs?

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 $\dim X\le 1$ or, equivalently, that $X$ is a dendrit.

8 pages