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

TL;DR

This paper studies quotient algebras of Banach operator ideals related to non-classical approximation properties, revealing conditions under which these algebras are trivial or contain complex ideal structures.

Contribution

It characterizes the nilpotent quotient algebras for specific operator ideals, providing bounds on their nilpotency index and constructing examples with intricate ideal chains.

Findings

If X has cotype 2, the quotient algebra is zero.

For p > 2, there exists a subspace with complex ideal chains.

The nilpotency index is bounded by a function of p.

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 $\mathfrak A_X^{\mathcal{QN}_p}$ and $\mathfrak A_X^{\mathcal{SK}_p}$ for the quasi $p$-nuclear operators $\mathcal{QN}_p$ and the Sinha-Karn $p$-compact operators $\mathcal{SK}_p$. The results include the following: (i) if $X$ has cotype 2, then $\mathfrak A_X^{\mathcal{QN}_p}=\{0\}$ for every $p\ge 1$; (ii) if $X^*$ has cotype 2, then $\mathfrak A_X^{\mathcal{SK}_p}=\{0\}$ for every $p\ge 1$; (iii) the exact upper bound of the index of nilpotency of $\mathfrak A_X^{\mathcal{QN}_p}$ and $\mathfrak A_X^{\mathcal{SK}_p}$ for $p\neq 2$ is $\max\{2,\left \lceil p/2 \right \rceil\}$, where $\left \lceil p/2 \right \rceil$ 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 $\mathfrak A_X^{\mathcal{QN}_p}$ and $\mathfrak A_X^{\mathcal{SK}_p}$ 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.