Ball proximinality of $M$-ideals of compact operators

TL;DR

This paper establishes the proximinality of the closed unit ball of $M$-ideals of compact operators and explores related properties of $M$-embedded spaces and specific operator spaces, revealing new proximinality results.

Contribution

It proves the ball proximinality of $M$-ideals of compact operators and $M$-embedded spaces, including the case of $ ext{K}( ext{l}_1)$ in $ ext{B}( ext{l}_1)$, even when not an $M$-ideal.

Findings

Closed unit ball of $M$-ideals of compact operators is proximinal.

$M$-embedded spaces are ball proximinal in their biduals.

$ ext{K}( ext{l}_1)$ is ball proximinal in $ ext{B}( ext{l}_1)$ despite not being an $M$-ideal.

Abstract

In this article, we prove the proximinality of closed unit ball of $M$-ideals of compact operators. We also prove the ball proximinality of $M$-embedded spaces in their biduals. Moreover, we show that $\mathcal{K}(\ell_1)$, the space of compact operators on $\ell_1$, is ball proximinal in $\mathcal{B}(\ell_1)$, the space of bounded operators on $\ell_1$, even though $\mathcal{K}(\ell_1)$ is not an $M$-ideal in $\mathcal{B}(\ell_1)$.