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

Abstract

A recent result of Leung (Proceedings of the American Mathematical Society, to appear) states that the Banach algebra $\mathcal{B}(X)$ of bounded, linear operators on the Banach space $X=({⨁}_{n\in \mathbb{N}}{\ell }_{\infty }^n{)}_{{\ell }_1}$ contains a unique maximal ideal. We show that the same conclusion holds true for the Banach spaces $X=({⨁}_{n\in \mathbb{N}}{\ell }_{\infty }^n{)}_{{\ell }_p}$ and $X=({⨁}_{n\in \mathbb{N}}{\ell }_1^n{)}_{{\ell }_p}$ whenever $p\in (1,\infty )$.

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• Closed ideals of operators on and complemented subspaces of Banach spaces of functions with countable support