The Bishop-Phelps-Bollob\'as version of Lindenstrauss properties A and B

TL;DR

This paper investigates a Bishop-Phelps-Bollobás version of Lindenstrauss properties A and B, establishing universal functions for these properties and characterizing spaces with these properties through isometric conditions and sum space analysis.

Contribution

It introduces a Bishop-Phelps-Bollobás framework for Lindenstrauss properties, providing necessary conditions, universal functions, and analyzing sum spaces to understand these properties.

Findings

Existence of universal functions for the BPBp in certain spaces

Necessary isometric conditions for spaces with the property

Characterization of spaces with the property via sum space analysis

Abstract

We study a Bishop-Phelps-Bollob\'as version of Lindenstrauss properties A and B. For domain spaces, we study Banach spaces $X$ such that $(X,Y)$ has the Bishop-Phelps-Bollob\'as property (BPBp) for every Banach space $Y$. We show that in this case, there exists a universal function $\eta_X(\varepsilon)$ such that for every $Y$, the pair $(X,Y)$ has the BPBp with this function. This allows us to prove some necessary isometric conditions for $X$ to have the property. We also prove that if $X$ has this property in every equivalent norm, then $X$ is one-dimensional. For range spaces, we study Banach spaces $Y$ such that $(X,Y)$ has the Bishop-Phelps-Bollob\'as property for every Banach space $X$. In this case, we show that there is a universal function $\eta_Y(\varepsilon)$ such that for every $X$, the pair $(X,Y)$ has the BPBp with this function. This implies that this property of $Y$ is strictly stronger than Lindenstrauss property B. The main tool to get these results is the study of the Bishop-Phelps-Bollob\'as property for $c_0$-, $\ell_1$- and $\ell_\infty$-sums of Banach spaces.