This is an announcement for the paper "The Bishop-Phelps-Bollob'as version of Lindenstrauss properties A" by Richard Aron, Yun Sung Choi, Sun Kwang Kim, Han Ju Lee, and Miguel Martin.

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 $\eta_X(\eps)$ 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.

Archive classification: math.FA

Mathematics Subject Classification: Primary 46B20, Secondary 46B04, 46B22

Submitted from: mmartins@ugr.es

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1305.6420

or

http://arXiv.org/abs/1305.6420