The Quotient Algebra of Compact-by-Approximable Operators on Banach Spaces Failing the Approximation Property

Abstract

We initiate a study of structural properties of the quotient algebra K(X)/A(X) of the compact-by-approximable operators on Banach spaces X failing the approximation property. Our main results and examples include the following: (i) there is a linear isomorphic embedding from c 0 into K(Z)/A(Z), where Z belongs to the class of Banach spaces constructed by Willis that have the metric compact approximation property but fail the approximation property, (ii) there is a linear isomorphic embedding from a non-separable space c 0 (Γ) into K(Z F J )/A(Z F J ), where Z F J is a universal compact factorisation space arising from the work of Johnson and Figiel.

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

“…However, by [53, Proposition 2.5] the embedding ψ cannot preserve any of the multiplicative structure of c 0 , and the construction does not ensure that A X is non-nilpotent. We next improve and complement these results from [53], and show there are also closed subspaces X ⊂ ℓ p , where 1 ≤ p < ∞ and p = 2, and X ⊂ c 0 , such that the quotient algebra A X is non-nilpotent and infinite-dimensional. These subspaces will also be used in later examples.…”

“…Nevertheless, these quotient algebras are natural examples of (typically) non-commutative radical Banach algebras, that is, the quotient elements S + A(X) are quasi-nilpotent for all S ∈ K(X). Recently various facts and problems about such algebras were highlighted by Dales [10], and his questions motivated the results and examples in [53] about the size of the quotient algebras A X for classes of Banach spaces X. In this paper we complement and expand our earlier study by looking more carefully at the algebraic structure of A X , in particular at examples of non-trivial closed two-sided ideals.…”

“…We showed in [53,Proposition 3.1. (i)] that if the Banach space X has the B.C.A.P., but fails the A.P., then there is a compact operator U ∈ K(X) such that U m / ∈ A(X) for any m ∈ N. In particular, the quotient algebra A X is non-nilpotent.…”

“…It follows from (2.3) that the quotient algebra A X is not nilpotent, since B 1 / ∈ A(X). We are now in position to repeat the argument of [53,Proposition 3.1]. In fact, A X is a nonnilpotent radical Banach algebra, which is infinite-dimensional by [9, Proposition 1.5.6.(iv)].…”

“…[16,Proposition 11.10]. Since A M = {0} and A N = {0}, the quotient algebras A X and A Y are both non-trivial by [53,Proposition 4.2]. Finally, since A X embeds isometrically into A X * by duality [53, Proposition 4.1], we also know that A X * = {0}.…”