Structure of closed subideals of $\mathcal L(X)$
Hans-Olav Tylli, Henrik Wirzenius
TL;DR
This work analyzes closed subideals of the operator algebra $\mathcal{L}(X)$ to reveal a richer and more nuanced ideal-structure than the classical lattice of closed ideals. It develops a general framework for closed $\mathcal{I}$-subideals, constructs explicit continuum families of nontrivial closed $\mathcal{K}(Z_p)$-subideals of $\mathcal{L}(Z_p)$, and demonstrates isomorphism patterns under natural identifications. It then extends these ideas to closed $n$-subideals (for $n\ge 2$), producing infinite decreasing chains of $(n+1)$-subideals that are not $n$-subideals in various spaces, including direct sums of $\ell^{p_j}$ and Tarbard-type HI spaces, with some subideals contained in $\mathcal{K}(X)$ when the approximation property fails. Overall, the paper shows that the closed subideal landscape of $\mathcal{L}(X)$ is rich and tightly connected to the geometry of $X$, the approximation property, and specialized constructions like Tarbard spaces and $Z_p$-type direct sums.
Abstract
The closed subalgebra $\mathcal J$ of the Banach algebra $\mathcal L(X)$ of bounded linear operators on the Banach space $X$ is a non-trivial closed $\mathcal I$-subideal of $\mathcal L(X)$ if $\mathcal I$ is a closed ideal of $\mathcal L(X)$ and $\mathcal J$ is an ideal of $\mathcal I$, but $\mathcal J$ is not an ideal of $\mathcal L(X)$. We obtain a variety of examples of non-trivial closed subideals of $\mathcal L(X)$ for different spaces $X$, which highlight further significant differences compared to the class of closed ideals. We study the concept of a closed $n$-subideal of $\mathcal L(X)$ for $n \ge 3$, which is a natural generalization of that of a closed subideal. In particular, we find explicit spaces $X$ for which $\mathcal L(X)$ contains a decreasing sequence $(\mathcal M_n)_{n\in \mathbb N}$ of closed subalgebras, where for all $n\in\mathbb N$ the subalgebra $\mathcal M_n$ is an $(n+1)$-subideal of $\mathcal L(X)$ but not an $n$-subideal. Moreover, we construct closed $n$-subideals contained in the compact operators $\mathcal K(X)$ for certain Banach spaces $X$ which fail the approximation property.