A non-linear Bishop-Phelps-Bollob\'as type theorem

TL;DR

This paper establishes a non-linear Bishop-Phelps-Bollobás type theorem for certain classes of uniform algebras on dual spaces, extending classical results to more general, non-linear settings.

Contribution

It proves a new Bishop-Phelps-Bollobás theorem for unital uniform algebras on dual spaces, including vector-valued cases, under specific convexity conditions.

Findings

The theorem holds for uniformly convex dual spaces.

It extends to vector-valued holomorphic functions.

Applicable to spaces with order continuous absolute norms.

Abstract

The main aim of this paper is to prove a Bishop-Phelps-Bollob\'as type theorem on the unital uniform algebra A_{w^*u}(B_{X^*}) consisting of all w^*-uniformly continuous functions on the closed unit ball B_{X^*} which are holomorphic on the interior of B_{X^*}. We show that this result holds for A_{w^*u}(B_{X^*}) if X^* is uniformly convex or X^* is the uniformly complex convex dual space of an order continuous absolute normed space. The vector-valued case is also studied.