Exotic closed subideals of algebras of bounded operators

Abstract

We exhibit a Banach space Z Z failing the approximation property, for which there is an uncountable family F \mathscr F of closed subideals contained in the Banach algebra K ( Z ) \mathcal K(Z) of the compact operators on Z Z , such that the subideals in F \mathscr F are mutually isomorphic as Banach algebras. This contrasts with the behaviour of closed ideals of the algebras L ( X ) \mathcal L(X) of bounded operators on X X , where closed ideals I ≠ J \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 S ( X ) \mathcal S(X) for classical spaces such as X = L p X = L^p with p ≠ 2 p \neq 2 , where pairwise isomorphic as well as pairwise non-isomorphic subideals occur.