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28 March 2024
 
  » arxiv » 1305.6420

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The Bishop-Phelps-Bollob'as version of Lindenstrauss properties A and B
Richard Aron ; Yun Sung Choi ; Sun Kwang Kim ; Han Ju Lee ; Miguel Martin ;
Date 28 May 2013
AbstractWe 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 $eta_X(eps)$ 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 $eta_Y(eps)$ 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.
Source arXiv, 1305.6420
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