Exotic closed subideals of algebras of bounded operators

Abstract

We exhibit a Banach space $Z$ failing the approximation property, for which there is an uncountable family $\mathcal{F}$ of closed subideals contained in the Banach algebra $\mathcal{K}(Z)$ of the compact operators on $Z$, such that the subideals in $\mathcal{F}$ 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 $\mathcal{I}\ne \mathcal{J}$ 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 $p\ne 2$, where pairwise isomorphic as well as pairwise non-isomorphic subideals occur.

15 pages; accepted to Proc. Amer. Math. Soc.; the main change to v1: Section 3 has been rewritten

References in corpus (3)

• Closed ideals in the algebra of compact-by-approximable operators
• Quotient algebra of compact-by-approximable operators on Banach spaces failing the approximation property
• Quotient algebras of Banach operator ideals related to non-classical approximation properties