Quotient algebra of compact-by-approximable operators on Banach spaces failing the approximation property

TL;DR

Addresses the structure of the quotient algebra $\mathfrak A_X=\mathcal K(X)/\mathcal A(X)$ for Banach spaces failing the approximation property. It develops general constructions showing $\mathfrak A_X$ is infinite-dimensional and provides explicit embeddings of $c_0$ into $\mathfrak A_Z$ for Willis spaces and non-separable $c_0(\Gamma)$ into $\mathfrak A_{Z^p_{FJ}}$, including duality considerations with $X^*$ and universal factorisation spaces. The findings reveal abundant nontrivial radical structure in these quotient algebras and offer concrete, computable examples across a range of universal constructions. They also discuss limitations of preserving algebraic structure in such embeddings and pose open questions about finite-dimensional cases and Pisier-type spaces.

Abstract

We initiate a study of structural properties of the quotient algebra $\mathcal K(X)/\mathcal A(X)$ 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 $\mathcal K(Z)/\mathcal A(Z)$, 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 $\mathcal K(Z_{FJ})/\mathcal A(Z_{FJ})$, where $Z_{FJ}$ is a universal compact factorisation space arising from the work of Johnson and Figiel.

Theorems & Definitions (31)

• Proposition 2.2
• proof
• Proposition 2.4
• proof
• Corollary 2.6
• Corollary 2.7
• proof
• Remark 2.8
• Proposition 2.9
• proof