Closed ideals in the algebra of compact-by-approximable operators

TL;DR

This paper constructs examples of non-trivial closed ideals in the algebra of compact-by-approximable operators on Banach spaces, revealing complex ideal structures and non-classical approximation properties.

Contribution

It provides the first known examples of non-trivial closed ideals in the algebra of compact-by-approximable operators on Banach spaces, especially those failing the approximation property.

Findings

Existence of at least two closed ideals in certain direct sums of Banach spaces.

Construction of subspaces with non-trivial closed ideals in $rak{A}_X$.

Presence of uncountably many closed ideals with complex order structure.

Abstract

We construct various examples of non-trivial closed ideals of the compact-by-approximable algebra $\mathfrak{A}_X =:\mathcal K(X)/\mathcal A(X)$ on Banach spaces $X$ failing the approximation property. The examples include the following: (i) if $X$ has cotype $2$, $Y$ has type $2$, $\mathfrak{A}_X \neq \{0\}$ and $\mathfrak{A}_Y \neq \{0\}$, then $\mathfrak{A}_{X \oplus Y}$ has at least $2$ closed ideals, (ii) there are closed subspaces $X \subset \ell^p$ for $4 < p < \infty$ and $X \subset c_0$ such that $\mathfrak{A}_X$ contains a non-trivial closed ideal, (iii) there is a Banach space $Z$ such that $\mathfrak{A}_Z$ contains an uncountable lattice of closed ideal having the reverse order structure of the power set of the natural numbers. Some of our examples involve non-classical approximation properties associated to various Banach operator ideals. We also discuss the existence of compact non-approximable operators $X \to Y$, where $X \subset \ell^p$ and $Y \subset \ell^q$ are closed subspaces for $p \neq q$.