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 \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. [ABSTRACT FROM AUTHOR]