Extension of linear operators and Lipschitz maps into C(K)-spaces

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 e T : X → C(K) with ‖e T‖ ≤ (1 + )‖T‖ (for any > 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 1 then every operator T : E → C(K) can be extended to an operator e T : 1 → C(K) with ‖e T‖ ≤ (1 + )‖T‖. We then show that 1 has a universal extension property: if X is a separable Banach space containing 1 then any operator T : 1 → C(K) can be extended to an operator e T : X → C(K) with ‖ e T‖ ≤ (1 + )‖T‖; this answers a question of Speegle.