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

TL;DR

This paper explores the structure of subspaces within ll_{} and characterizes extreme contractions in spaces of bounded linear operators, especially focusing on finite-dimensional polyhedral spaces and their ll_{}^n subspaces.

Contribution

It provides new insights into the polyhedrality of subspaces of ll_{} and explicitly computes the number of extreme contractions in ll(,^n) for finite-dimensional polyhedral spaces.

Findings

Characterization of polyhedral subspaces of ll_{}

Explicit count of extreme contractions in ll(,^n)

Results applicable to finite-dimensional polyhedral spaces

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 \geq 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.