Which compacta are noncommutative ARs?

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