Simultaneously continuous retraction and Bishop-Phelps-Bollobás type theorem

The dual space X^* of a Banach space X is said to admit a uniformly simultaneously continuous retraction if there is a retraction r from X^* onto its unit ball BX* which is uniformly continuous in norm topology and continuous in weak-* topology. We prove that if a Banach space (resp. complex Banach space) X has a normalized unconditional Schauder basis with unconditional basis constant 1 and if X^* is uniformly monotone (resp. uniformly complex convex), then X^* admits a uniformly simultaneously continuous retraction. It is also shown that X^* admits such a retraction if X=[{N-ary circled plus operator}Xi]c0 or X=[{N-ary circled plus operator}Xi]ℓ1, where {X_i} is a family of separable Banach spaces whose duals are uniformly convex with moduli of convexity δ_i(ε) with inf_iδ_i(ε)>0 for all 0<ε<1. Let K be a locally compact Hausdorff space and let C₀(K) be the real Banach space consisting of all real-valued continuous functions vanishing at infinity. As an application of simultaneously continuous retractions, we show that a pair (X, C₀(K)) has the Bishop-Phelps-Bollobás property for operators if X^* admits a uniformly simultaneously continuous retraction. As a corollary, (C₀(S), C₀(K)) has the Bishop-Phelps-Bollobás property for operators for every locally compact metric space S.