Operator Space Structures on $\ell^1(n).$

TL;DR

This paper investigates the operator space structures of finite-dimensional $ ext{ell}^1(n)$ spaces, proving they cannot be isometrically embedded into matrices and exploring infinite-dimensional structures.

Contribution

It establishes the non-existence of isometric embeddings of $ ext{ell}^1(n)$ into matrix spaces and discusses new infinite-dimensional operator space structures.

Findings

No isometric embedding of $ ext{ell}^1(n)$ into $k imes k$ matrices for any $k$.

Discussion of a class of infinite-dimensional operator space structures on $ ext{ell}^1(n)$.

Abstract

We show that the complex normed linear space $\ell^1(n),$ $n>1,$ has no isometric embedding into $k\times k$ complex matrices for any $k\in \mathbb{N}$ and discuss a class of infinite dimensional operator space structures on it.