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arXiv · 2401.16135

The Bishop-Phelps-Bollobas property for certain Banach spaces

Tirthankar Bhattacharyya·Mainak Bhowmik·Kousik Dhara
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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-Bollob\'as 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.

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Tirthankar Bhattacharyya, Mainak Bhowmik, Kousik Dhara. 2024-01-29. The Bishop-Phelps-Bollobas property for certain Banach spaces. https://arxiv.org/abs/2401.16135

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