Exotic closed subideals of algebras of bounded operators

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 $\mathcal{I}\mathrm{\ne }\mathcal{J}$ 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 $p\mathrm{\ne }2$, where pairwise isomorphic as well as pairwise non-isomorphic subideals occur.