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 F of closed subideals contained in the Banach algebra K(Z) of the compact operators on Z, such that the subideals in F are mutually isomorphic as Banach algebras. This contrasts with the behaviour of closed ideals of the algebras L(X) of bounded operators on X, where closed ideals I = J are never isomorphic as Banach algebras. We also construct families of non-trivial closed subideals in the strictly singular operators S(X) for the classical spaces X = L p with p = 2, where the subideals are not pairwise isomorphic.