The Bishop-Phelps-Bollobas property for certain Banach spaces

Abstract

Let $X$ be a complex Banach space. We prove that if $L$ is an extremally disconnected compact Hausdorff topological space, then the pair $(X,C(L))$ satisfies the Bishop-Phelps-Bollobás property (BPBp for short). As a byproduct, we obtain the BPBp for the pair $(X,L^{\infty }(\nu ))$ for any measure $\nu$. In particular, this settles an unresolved question regarding the BPBp for the pair $(L^{\infty }(\mu ),L^{\infty }(\nu ))$ for any two measures $\mu$ and $\nu$. Finally, we show that $(X,H^{\infty }(\Omega )$ has the BPBp when $\Omega$ is a multi-connected planar domain bounded by finitely many disjoint analytic simple closed curves.

There is an error in Lemma 3.1. So we withdraw the paper