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 C1 of all (not necessarily commutative) unital C*-algebras if and only if X is an absolute retract of dimension dimX⩽1 or, equivalently, that X is a dendrit.