Exotic closed subideals of algebras of bounded operators

TL;DR

This paper constructs examples of Banach spaces with uncountably many mutually isomorphic closed subideals of compact operators, contrasting with the uniqueness of ideal structures in classical operator algebras.

Contribution

It demonstrates the existence of uncountably many mutually isomorphic closed subideals in the algebra of compact operators on a Banach space lacking the approximation property, revealing new structural phenomena.

Findings

Existence of uncountably many mutually isomorphic closed subideals in $\,\mathcal K(Z)$

Construction of non-trivial closed subideals in strictly singular operators on classical spaces

Contrast with the behavior of closed ideals in $\,\mathcal L(X)$

Abstract

We exhibit a Banach space $Z$ failing the approximation property, for which there is an uncountable family $\mathscr F$ of closed subideals contained in the Banach algebra $\mathcal K(Z)$ of the compact operators on $Z$, such that the subideals in $\mathscr F$ are mutually isomorphic as Banach algebras. This contrasts with the behaviour of closed ideals of the algebras $\mathcal L(X)$ of bounded operators on $X$, where closed ideals $\mathcal I \neq \mathcal J$ are never isomorphic as Banach algebras. We also construct families of non-trivial closed subideals contained in the strictly singular operators $\mathcal S(X)$ for classical spaces such as $X = L^p$ with $p \neq 2$, where pairwise isomorphic as well as pairwise non-isomorphic subideals occur.