Simultaneously continuous retraction and Bishop-Phelps-Bollobas type theorem

Abstract

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 = [circle plus X-i](c0) or X = [circle plus X-i](l1), where {X-i} is a family of separable Banach spaces whose duals are uniformly convex with moduli of convexity delta(i)(epsilon) with inf(i) delta(i)(epsilon) > 0 for all 0 < epsilon < 1. Let K be a locally compact Hausdorff space and let (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. (C) 2014 Elsevier Inc. All rights reserved.