Ball proximinality of $M$-ideals of compact operators

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 $\mathcal{B}({\ell }_1)$.