Simultaneously continuous retraction and Bishop-Phelps-Bollobás type theorem

Abstract

We study the existence of a retraction from the dual space $X^{\ast }$ of a (real or complex) Banach space $X$ onto its unit ball $B_{X^{\ast }}$ which is uniformly continuous in norm topology and continuous in weak-$\ast$ 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 $X^{\ast }$ is uniformly monotone, then a uniformly simultaneously continuous retraction from $X^{\ast }$ onto $B_{X^{\ast }}$ exists. It is also shown that if $\{X_i\}$ is a family of separable Banach spaces whose duals are uniformly convex with moduli of convexity ${\delta }_i(\epsilon )$ such that ${\inf }_i{\delta }_i(\epsilon )>0$ and $X={\left[⨁X_i\right]}_{c_0}$ or $X={\left[⨁X_i\right]}_{{\ell }_p}$ for $1\le p<\infty$, then a uniformly simultaneously continuous retraction exists from $X^{\ast }$ onto $B_{X^{\ast }}$. The relation between the existence of a uniformly simultaneously continuous retraction and the Bishsop-Phelps-Bollobás property for operators is investigated and it is proved that the existence of a uniformly simultaneously continuous retraction from $X^{\ast }$ onto its unit ball implies that a pair $(X,C_0(K))$ has the Bishop-Phelps-Bollobás property for every locally compact Hausdorff spaces $K$. As a corollary, we prove that $(C_0(S),C_0(K))$ has the Bishop-Phelps-Bollobás property if $C_0(S)$ and $C_0(K)$ 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.

15 pages