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

Abstract

We construct various examples of non-trivial closed ideals of the compact-by-approximable algebra ${\mathfrak{A}}_X=:\mathcal{K}(X)/\mathcal{A}(X)$ on Banach spaces $X$ failing the approximation property. The examples include the following: (i) if $X$ has cotype $2$, $Y$ has type $2$, ${\mathfrak{A}}_X\ne \{0\}$ and ${\mathfrak{A}}_Y\ne \{0\}$, then ${\mathfrak{A}}_{X\oplus Y}$ has at least $2$ closed ideals, (ii) there are closed subspaces $X\subset {\ell }^p$ for $4<p<\infty$ and $X\subset c_0$ such that ${\mathfrak{A}}_X$ contains a non-trivial closed ideal, (iii) there is a Banach space $Z$ such that ${\mathfrak{A}}_Z$ 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 $X\to Y$, where $X\subset {\ell }^p$ and $Y\subset {\ell }^q$ are closed subspaces for $p\ne q$.

arXiv comment37 pages

References in corpus (3)

• The Compact Approximation Property does not imply the Approximation Property
• Hereditarily indecomposable, separable L_\infty spaces with \ell_1 dual having few operators, but not very few operators
• Quotient algebra of compact-by-approximable operators on Banach spaces failing the approximation property

Cited by in corpus (3)

• Quotient algebras of Banach operator ideals related to non-classical approximation properties
• On approximation properties related to unconditionally p-compact operators and Sinha-Karn p-compact operators
• Exotic closed subideals of algebras of bounded operators