We study the existence of a retraction from the dual space Xβ of a (real
or complex) Banach space X onto its unit ball BXββ which is uniformly
continuous in norm topology and continuous in weak-β topology. Such a
retraction is called a uniformly simultaneously continuous retraction.
It is shown that if X has a normalized unconditional Schauder basis with
unconditional basis constant 1 and Xβ is uniformly monotone, then a
uniformly simultaneously continuous retraction from Xβ onto BXββ
exists. It is also shown that if {Xiβ} is a family of separable Banach
spaces whose duals are uniformly convex with moduli of convexity
Ξ΄iβ(Ξ΅) such that infiβΞ΄iβ(Ξ΅)>0 and X=[β¨Xiβ]c0ββ or X=[β¨Xiβ]βpββ
for 1β€p<β, then a uniformly simultaneously continuous retraction
exists from Xβ onto BXββ.
The relation between the existence of a uniformly simultaneously continuous
retraction and the Bishsop-Phelps-Bollob\'as property for operators is
investigated and it is proved that the existence of a uniformly simultaneously
continuous retraction from Xβ onto its unit ball implies that a pair (X,C0β(K)) has the Bishop-Phelps-Bollob\'as property for every locally compact
Hausdorff spaces K. As a corollary, we prove that (C0β(S),C0β(K)) has the
Bishop-Phelps-Bollob\'as property if C0β(S) and C0β(K) are the spaces of
all real-valued continuous functions vanishing at infinity on locally compact
metric space S and locally compact Hausdorff space K respectively.Comment: 15 page