Bishop-Phelps-Bollob\'as property for positive operators when the domain is $C_0(L) $

María D. Acosta; Maryam Soleimani-Mourchehkhorti

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arXiv · 2108.01638

Bishop-Phelps-Bollob\'as property for positive operators when the domain is $C_0(L) $

María D. Acosta·Maryam Soleimani-Mourchehkhorti

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Abstract

Recently it was introduced the so-called Bishop-Phelps-Bollob{\'a}s property for positive operators between Banach lattices. In this paper we prove that the pair $(C_0(L), Y) $ has the Bishop-Phelps--Bollob{\'a}s property for positive operators, for any locally compact Hausdorff topological space $L$, whenever $Y$ is a uniformly monotone Banach lattice with a weak unit. In case that the space $C_0(L)$ is separable, the same statement holds for any uniformly monotone Banach lattice $Y .$ We also show the following partial converse of the main result. In case that $Y$ is a strictly monotone Banach lattice, $L$ is a locally compact Hausdorff topological space that contains at least two elements and the pair $(C_0(L), Y )$ has the Bishop-Phelps--Bollob{\'a}s property for positive operators then $Y$ is uniformly monotone.

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María D. Acosta, Maryam Soleimani-Mourchehkhorti. 2021-08-03. Bishop-Phelps-Bollob\'as property for positive operators when the domain is $C_0(L) $. https://arxiv.org/abs/2108.01638

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