Uniqueness of the maximal ideal of operators on the $\ell_p$-sum of $\ell_\infty^n\ (n\in\mathbb{N})$ for $1<p<\infty$

TL;DR

This paper extends the known uniqueness of the maximal ideal in the algebra of bounded operators to Banach spaces formed by $igl(igoplus_{n oig)} ext{l}_ ext{infty}^n$ and $ ext{l}_1^n$ sums with $1<p< ext{infty}$, broadening previous results.

Contribution

It proves the maximal ideal's uniqueness for operator algebras on $ ext{l}_p$-sums of $ ext{l}_ extinfty^n$ and $ ext{l}_1^n$ spaces, generalizing Leung's result.

Findings

The maximal ideal in the algebra of bounded operators is unique for these Banach spaces.

The result applies to spaces formed by $ ext{l}_p$-sums with $1<p< extinfty$.

The conclusion extends previous work on similar Banach spaces.

Abstract

A recent result of Leung (Proceedings of the American Mathematical Society, to appear) states that the Banach algebra $\mathscr{B}(X)$ of bounded, linear operators on the Banach space $X=\bigl(\bigoplus_{n\in\mathbb{N}}\ell_\infty^n\bigr)_{\ell_1}$ contains a unique maximal ideal. We show that the same conclusion holds true for the Banach spaces $X=\bigl(\bigoplus_{n\in\mathbb{N}}\ell_\infty^n\bigr)_{\ell_p}$ and $X=\bigl(\bigoplus_{n\in\mathbb{N}}\ell_1^n\bigr)_{\ell_p}$ whenever $p\in(1,\infty)$.