Extension of operators on spaces of vector-valued continuous functions

Abstract

Let X be a completely regular Hausdorff space and let Ba stand for the Baire σ-algebra of sets in X. For a Banach space E let Crc(X,E) be the space of all continuous functions f:X→E such that f(X) is a relatively compact set in E, and let B(Ba,E) be the space of all totally Ba-measurable functions f:X→E. For a Banach space Y we study the problem of extension of some natural classes of bounded linear operators T:Crc(X,E)→Y to operators T¯:B(Ba,E)→Y.