Complemented ideals of ℓ∞

Answering questions raised by Leonetti and by Rincón-Villamizar and Uzcátegui-Aylwin we characterize ideals I⊆P(ω) such that c0,I is complemented in ℓ∞ as exactly those ideals for which the space KI=Stone(P(ω)/I) is approximable, i.e., the unit ball of the space M(KI) of signed Radon measures on KI is separable in the weak∗ topology.