Minimal and maximal $p$-operator space structures

TL;DR

This paper investigates the minimal and maximal $p$-operator space structures of $L^ Infty(1)$ and $L^1(1)$, revealing their properties as multiplication operators and tensor products, respectively.

Contribution

It establishes that $L^1(1)$ has a maximal $p$-operator space structure, and shows $L^1(1)$'s structure facilitates tensor product computations.

Findings

$L^1(1)$ has a maximal $p$-operator space structure.

$L^1(1)$ as multiplication operators on $L^p(1)$ are minimal.

Results aid in tensor product calculations involving $L^1(1)$.

Abstract

We show that $L^\infty(\mu)$, in its capacity as multiplication operators on $L^p(\mu)$, is minimal as a $p$-operator space for a decomposable measure $\mu$. We conclude that $L^1(\mu)$ has a certain maximal type $p$-operator space structure which facilitates computations with $L^1(\mu)$ and the projective tensor product.