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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]

We construct various examples of non-trivial closed ideals of the compact-by-approximable algebra A X : = K (X) / A (X) on Banach spaces X failing the approximation property. The examples include the following: (i) if X has cotype 2, Y has type 2, A X ≠ { 0 } and A Y ≠ { 0 } , then A X ⊕ Y has at least 2 closed ideals, (ii) there are closed subspaces X ⊂ ℓ p for 4 < p < ∞ and X ⊂ c 0 such that A X contains a non-trivial closed ideal, (iii) there is a Banach space Z such that A Z contains an uncountable lattice of closed ideals having the reverse order structure of the power set of the natural numbers. Some of our results involve non-classical approximation properties associated to various Banach operator ideals. We also discuss the existence of compact non-approximable operators X → Y , where X ⊂ ℓ p and Y ⊂ ℓ q are closed subspaces for p ≠ q. [ABSTRACT FROM AUTHOR]