Quotient algebras of Banach operator ideals related to non-classical approximation properties

Abstract

We investigate the quotient algebra ${\mathfrak{A}}_X^{\mathcal{I}}:=\mathcal{I}(X)/{\overline{\mathcal{F}(X)}}^{||\cdot |{|}_{\mathcal{I}}}$ for Banach operator ideals $\mathcal{I}$ contained in the ideal of the compact operators, where $X$ is a Banach space that fails the $\mathcal{I}$-approximation property. The main results concern the nilpotent quotient algebras ${\mathfrak{A}}_X^{{\mathcal{Q}\mathcal{N}}_p}$ and ${\mathfrak{A}}_X^{{\mathcal{S}\mathcal{K}}_p}$ for the quasi $p$-nuclear operators ${\mathcal{Q}\mathcal{N}}_p$ and the Sinha-Karn $p$-compact operators ${\mathcal{S}\mathcal{K}}_p$. The results include the following: (i) if $X$ has cotype 2, then ${\mathfrak{A}}_X^{{\mathcal{Q}\mathcal{N}}_p}=\{0\}$ for every $p\ge 1$; (ii) if $X^{\ast }$ has cotype 2, then ${\mathfrak{A}}_X^{{\mathcal{S}\mathcal{K}}_p}=\{0\}$ for every $p\ge 1$; (iii) the exact upper bound of the index of nilpotency of ${\mathfrak{A}}_X^{{\mathcal{Q}\mathcal{N}}_p}$ and ${\mathfrak{A}}_X^{{\mathcal{S}\mathcal{K}}_p}$ for $p\ne 2$ is $\max \{2,\lceil p/2\rceil \}$, where $\lceil p/2\rceil$ denotes the smallest $n\in \mathbb{N}$ such that $n\ge p/2$; (iv) for every $p>2$ there is a closed subspace $X\subset c_0$ such that both ${\mathfrak{A}}_X^{{\mathcal{Q}\mathcal{N}}_p}$ and ${\mathfrak{A}}_X^{{\mathcal{S}\mathcal{K}}_p}$ contain a countably infinite decreasing chain of closed ideals. In addition, our methods yield a closed subspace $X\subset c_0$ such that the compact-by-approximable algebra ${\mathfrak{A}}_X=\mathcal{K}(X)/\mathcal{A}(X)$ contains two incomparable countably infinite chains of nilpotent closed ideals.

35 pages

References in corpus (2)

• Closed ideals in the algebra of compact-by-approximable operators
• Quotient algebra of compact-by-approximable operators on Banach spaces failing the approximation property

Cited by in corpus (2)

• On approximation properties related to unconditionally p-compact operators and Sinha-Karn p-compact operators
• Exotic closed subideals of algebras of bounded operators