Simultaneously continuous retraction and Bishop-Phelps-Bollob\'as type theorem

TL;DR

This paper investigates the existence of a special type of retraction from the dual space of a Banach space onto its unit ball, exploring conditions under which such retractions exist and their implications for the Bishop-Phelps-Bollobás property.

Contribution

It establishes new conditions for the existence of uniformly simultaneously continuous retractions in Banach spaces and links these to the Bishop-Phelps-Bollobás property for operator pairs.

Findings

Existence of such retractions under unconditional basis and uniform monotonicity.

Retractions exist for certain direct sums of Banach spaces with uniformly convex duals.

Presence of retractions implies the Bishop-Phelps-Bollobás property for pairs involving $C_0$ spaces.

Abstract

We study the existence of a retraction from the dual space $X^*$ of a (real or complex) Banach space $X$ onto its unit ball $B_{X^*}$ which is uniformly continuous in norm topology and continuous in weak-$*$ 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 $X^*$ is uniformly monotone, then a uniformly simultaneously continuous retraction from $X^*$ onto $B_{X^*}$ 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 $\delta_i(\varepsilon)$ such that $\inf_i \delta_i(\varepsilon)>0$ and $X= \left[\bigoplus X_i\right]_{c_0}$ or $X=\left[\bigoplus X_i\right]_{\ell_p}$ for $1\le p<\infty$, then a uniformly simultaneously continuous retraction exists from $X^*$ onto $B_{X^*}$. 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 $X^*$ onto its unit ball implies that a pair $(X, C_0(K))$ has the Bishop-Phelps-Bollob\'as property for every locally compact Hausdorff spaces $K$. As a corollary, we prove that $(C_0(S), C_0(K))$ has the Bishop-Phelps-Bollob\'as 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.