We study the extension of linear operators with range in a C(K)-space, comparing and contrasting our results with the corresponding results for the nonlinear problem of extending Lipschitz maps with values in a C(K)-space. We give necessary and sufficient conditions on a separable Banach space X which ensure that every operator T:E→C(K) defined on a subspace may be extended to an operator \tilde T:X→C(K) with ∥\tilde T∥≦ (1+ε)∥T∥ (for any ε>0). Based on these we give new examples of such spaces (including all Orlicz sequence spaces with separable dual for a certain equivalent norm). We answer a question of Johnson and Zippin by showing that if E is a weak*-closed subspace of ℓ₁ then every operator T:E→C(K) can be extended to an operator \tilde T:ℓ₁→C(K) with ∥\tilde T∥≦ (1+ε)∥T∥. We then show that ℓ₁ has a universal extension property: if X is a separable Banach space containing ℓ₁ then any operator T:ℓ₁→C(K) can be extended to an operator \tilde T:X→ C(K) with ∥\tilde T∥≦ (1+ε)∥T∥; this answers a question of Speegle.

The author was supported by NSF grant DMS-0555670