Abstract
We construct various examples of non-trivial closed ideals of the compact-by-approximable 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⊂ℓp 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 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⊂ℓp and Y⊂ℓq are closed subspaces for p≠q.
Highlights
Given Banach spaces X and Y, let K(X, Y ) be the class of compact operators X → Y
Recall that Banach spaces X failing the approximation property are complicated objects to construct or recognise
In Theorem 3.14 we will construct a direct sum X ⊕ Y from this class of spaces for which K(X ⊕ Y ), and AX⊕Y, contains at least 8 non-trivial closed ideals. Such examples require more preparation, including a study of the strict inclusion A(X, Y ) K(X, Y ) among closed subspaces X ⊂ p and Y ⊂ q for p = q, as well as non-classical approximation properties associated to Banach operator ideals
Summary
Given Banach spaces X and Y , let K(X, Y ) be the class of compact operators X → Y. In Theorem 2.9 we find closed subspaces X ⊂ p for p ∈ [1, ∞) and p = 2, as well as X ⊂ c0, for which the quotient algebra AX is nonnilpotent and infinite-dimensional This result improves on [57, section 2], and it will be crucial in some of our later examples of closed ideals of the compact-by-approximable algebra. The fact that this holds in the real case is a consequence of classical Riesz-Fredholm theory, which does not depend of the scalar field (see Proposition 2.8 for the precise details) With this understanding our results and examples are independent of the scalar field R or C of the underlying Banach space X. Recall from [10, Question 2.2.A, page 182] that it is a longstanding open problem whether there are topologically simple radical Banach algebras A, that is, A has a non-trivial product and no non-trivial closed ideals
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