Closed ideals in the algebra of compact-by-approximable operators

Abstract

We construct various examples of non-trivial closed ideals of the compact-byapproximable algebra AX =: 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, AX = {0} and AY = {0}, then AX⊕Y has at least 2 closed ideals, (ii) there are closed subspaces X ⊂ ℓ p for 4 < p < ∞ and X ⊂ c0 such that AX contains a non-trivial closed ideal, (iii) there is a Banach space Z such that AZ contains an uncountable lattice of closed ideal having the reverse order structure of the power set of the natural numbers. Some of our examples 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.