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=[⨁Xi]c0 or X=[⨁Xi]ℓ1, where {Xi} is a family of separable Banach spaces whose duals are uniformly convex with moduli of convexity δi(ε) with infiδi(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.