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

Tomasz Kania, Niels Jakob Laustsen|arXiv (Cornell University)|May 22, 2014

Advanced Banach Space Theory11 references3 citations

TL;DR

This paper establishes that the Banach algebras of bounded linear operators on the $σ$-finite $σ$-sum spaces $W_p = \left(\bigoplus_{n\in\mathbb{N}} \ell^n_\infty\right)_{\ell^p}$ and their duals $W_p^* = \left(\bigoplus_{n\in\mathbb{N}} \ell^n_1\right)_{\ell^q}$, for $1 < p < \infty$, possess a unique maximal ideal. The proof hinges on showing that the set of operators not factoring the identity through them coincides with a closed operator ideal of operators uniformly fixing the family $\{\ell^n_\infty\}$, using ultraproduct techniques and properties of strictly singular operators on minimal 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 $p\in(1,\infty)$.

Motivation & Objective

• To extend Leung's result on the unique maximal ideal in $B(W_1)$ to the full range $1 < p < \infty$ for the spaces $W_p = \left(\bigoplus_{n\in\mathbb{N}} \ell^n_\infty\right)_{\ell^p}$.
• To establish the same uniqueness result for the dual spaces $W_p^* = \left(\bigoplus_{n\in\mathbb{N}} \ell^n_1\right)_{\ell^q}$, where $q$ is the conjugate exponent of $p$.
• To show that the set $M_X = \{T \in B(X) : \text{identity on } X \text{ does not factor through } T\}$ is a closed operator ideal for $X = W_p$ and $X = W_p^*$, thereby proving it is the unique maximal ideal.
• To introduce and analyze a new operator ideal $S_{\{\ell^n_p : n\in\mathbb{N}\}}(X,Y)$ of operators that do not uniformly fix the family $\{\ell^n_p\}$, and prove its closedness for $p \in [1,\infty]$.
• To demonstrate that for $p \in (1,\infty)$, the set $M_{W_p}$ coincides with the operator ideal $S_{\{\ell^n_\infty : n\in\mathbb{N}\}}(W_p)$, which is key to proving uniqueness of the maximal ideal.

Proposed method

• Introduces a new operator ideal $S_{\{\ell^n_p : n\in\mathbb{N}\}}(X,Y)$ as the set of operators $T \in B(X,Y)$ that do not uniformly fix the family $\{\ell^n_p\}$, defined via uniform $C$-fixing of copies of $\ell^n_p$.
• Proves that $S_{\{\ell^n_p : n\in\mathbb{N}\}}$ forms a closed operator ideal in the sense of Pietsch for all $p \in [1,\infty]$, using the minimality of $\ell^p$-spaces and properties of strictly singular operators.
• Employs ultraproduct techniques to analyze the behavior of operators on $W_p$ and $W_p^*$, particularly in relation to factorization of the identity operator.
• Uses the fact that $W_p$ is reflexive for $p \in (1,\infty)$ to transfer the ideal structure from $B(W_p)$ to $B(W_p^*)$ via the adjoint map $T \mapsto T^*$, which induces an order isomorphism on the lattice of ideals.
• Applies Lemma 2.1 on perturbations of bounded below operators to show that if an operator $T$ $C$-fixes a copy of $\ell^n_\infty$, then nearby operators $S$ with $\|S - T\|$ small enough also $C'$-fix it for $C' > C$, ensuring stability of the fixed copy property.
• Constructs a sequence of finite-rank approximations and uses commutative diagrams involving projections $P_k$, $P_{k'}$, and $P_m$ to build a pair of operators $R$ and $S$ such that $ST'R = I_{W_p}$, proving that $T$ factors the identity if it uniformly fixes $\ell^n_\infty$.

Experimental results

Research questions

• RQ1 Does the Banach algebra $B(W_p)$ for $1 < p < \infty$ have a unique maximal ideal, extending Leung's result from $p=1$?
• RQ2 Is the set $M_{W_p} = \{T \in B(W_p) : \text{identity on } W_p \text{ does not factor through } T\}$ a closed ideal in $B(W_p)$, and if so, is it the unique maximal ideal?
• RQ3 Can the set $M_{W_p}$ be characterized as the operator ideal $S_{\{\ell^n_\infty : n\in\mathbb{N}\}}(W_p)$ of operators that do not uniformly fix the family $\{\ell^n_\infty\}$?
• RQ4 Does the same uniqueness result hold for the dual space $W_p^* = \left(\bigoplus_{n\in\mathbb{N}} \ell^n_1\right)_{\ell^q}$, where $q$ is the conjugate exponent of $p$?
• RQ5 Is the operator ideal $S_{\{\ell^n_p : n\in\mathbb{N}\}}$ closed under addition and thus forms a proper operator ideal in the sense of Pietsch for $p \in [1,\infty]$?

Key findings

• For each $p \in (1,\infty)$, the set $M_{W_p} = \{T \in B(W_p) : \text{identity on } W_p \text{ does not factor through } T\}$ is a closed ideal and is the unique maximal ideal of $B(W_p)$, as established by showing $M_{W_p} = S_{\{\ell^n_\infty : n\in\mathbb{N}\}}(W_p)$.
• The set $M_{W_p^*}$ is the unique maximal ideal of $B(W_p^*)$, obtained via the adjoint map $T \mapsto T^*$, which induces an order isomorphism between the lattices of ideals of $B(W_p)$ and $B(W_p^*)$, and maps $M_{W_p}$ to $M_{W_p^*}$.
• The operator ideal $S_{\{\ell^n_p : n\in\mathbb{N}\}}(X,Y)$ is closed under addition and thus forms a closed operator ideal in the sense of Pietsch for all $p \in [1,\infty]$, due to the minimality of $\ell^p$-spaces and the use of ultraproduct techniques.
• An operator $T \in B(W_p)$ uniformly fixes the family $\{\ell^n_\infty\}$ if and only if the identity on $W_p$ factors through $T$, which is the key equivalence used to identify $M_{W_p}$ with the operator ideal $S_{\{\ell^n_\infty : n\in\mathbb{N}\}}(W_p)$.
• The construction of operators $R$ and $S$ such that $ST'R = I_{W_p}$, using finite-rank approximations and projections, proves that if $T$ uniformly fixes $\ell^n_\infty$, then the identity factors through $T$, completing the characterization.
• The proof relies on ultraproducts to analyze the behavior of operators on $W_p$, particularly in showing that finite-rank perturbations of $T$ in the ultrapower preserve the property of fixing $c_0$ or $\ell^n_\infty$, which is essential for the factorization argument.