  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.
