SIMULTANEOUSLY CONTINUOUS RETRACTION AND BISHOP-PHELPS-BOLLOBÁS TYPE THEOREM

Abstract. The dual space X ∗ of a Banach space X is said to admit a uniformly simul-taneously 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 an retraction if X =