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 B X ⁎ 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 = [ ⨁ X i ] c 0 or X = [ ⨁ X i ] l 1 , where { X i } is a family of separable Banach spaces whose duals are uniformly convex with moduli of convexity δ i ( e ) with inf i δ i ( e ) > 0 for all 0 e 1 . Let K be a locally compact Hausdorff space and let C 0 ( 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 , C 0 ( K ) ) has the Bishop–Phelps–Bollobas property for operators if X ⁎ admits a uniformly simultaneously continuous retraction. As a corollary, ( C 0 ( S ) , C 0 ( K ) ) has the Bishop–Phelps–Bollobas property for operators for every locally compact metric space S.