The Bishop-Phelps-Bollobás version of Lindenstrauss properties A and B

Abstract

We study a Bishop-Phelps-Bollobás version of Lindenstrauss properties A and B. For domain spaces, we study Banach spaces $X$ such that $(X,Y)$ has the Bishop-Phelps-Bollobás property (BPBp) for every Banach space $Y$. We show that in this case, there exists a universal function ${\eta }_X(\epsilon )$ 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-Bollobás property for every Banach space $X$. In this case, we show that there is a universal function ${\eta }_Y(\epsilon )$ 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-Bollobás property for $c_0$-, ${\ell }_1$- and ${\ell }_{\infty }$-sums of Banach spaces.

Minor changes; accepted for publication in Trans. Amer. Math. Soc

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