A Bishop-Phelps-Bollobas Type Theorem for uniform algegras

Abstract

This paper is devoted to showing that Asplund operators with range in a uniform Banach algebra have the Bishop¿Phelps¿Bollobas property, i.e., they are approximated by norm attaining Asplund operators at the same time that a point where the approximated operator almost attains its norm is approximated by a point at which the approximating operator attains it. To prove this result we use the weak*-to-norm fragmentability of weak*-compact subsets of the dual of Asplund spaces and we need to observe a Urysohn type result producing peak complex-valued functions in uniform algebras that are small outside a given open set and whose image is inside a Stolz region.