On subspaces of ${\ell }_{\infty }$ and extreme contraction in $\mathbb{L}(\mathbb{X},{\ell }_{\infty }^n)$

Abstract

We investigate different possiblities of subspaces of the space ${\ell }_{\infty }$ in terms of whether the subspaces are polyhedral or not. We further study finite-dimensional subspaces of ${\ell }_{\infty }$ which are of the form ${\ell }_{\infty }^n$ form some $n\ge 2.$ As an application of the results we compute the number of extreme contractions for a class of the space of bounded linear operators. In particular we find the number of extreme contractions of $\mathbb{L}(\mathbb{X},{\ell }_{\infty }^n),$ where $\mathbb{X}$ is a finite-dimensional polyhedral space.