New York Journal of Mathematics Volume 13 (2007) 317-381
 Volume 13

N. J. Kalton

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

 Published: September 10, 2007 Keywords: Banach spaces, spaces of continuous functions, Lipschitz extensions, linear extensions Subject: Primary: 46B03, 46B20

Abstract
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:ℓ1C(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:ℓ1C(K) can be extended to an operator \tilde T:X→ C(K) with ∥\tilde T∥≦ (1+ε)∥T∥; this answers a question of Speegle.

Acknowledgements

The author was supported by NSF grant DMS-0555670

Author information

Department of Mathematics, University of Missouri-Columbia, Columbia, MO 65211
nigel@math.missouri.edu