On the Bounded Approximation Property in Banach spaces

TL;DR

This paper proves that kernels of quotient operators from $\\mathcal{L}_1$-spaces onto Banach spaces with the BAP also have the BAP, extending previous results and establishing stability under various quotient mappings.

Contribution

It establishes the BAP for kernels of quotient operators from $\\mathcal{L}_1$-spaces onto Banach spaces with BAP, generalizing earlier specific cases.

Findings

Kernels of quotient maps from $\\mathcal{L}_1$-spaces onto BAP spaces have BAP.

The BAP property is stable under various quotient mappings.

Dual results hold for $\\mathcal{L}_ty$-spaces.

Abstract

We prove that the kernel of a quotient operator from an $\mathcal L_1$-space onto a Banach space $X$ with the Bounded Approximation Property (BAP) has the BAP. This completes earlier results of Lusky --case $\ell_1$-- and Figiel, Johnson and Pe\l czy\'nski --case $X^*$ separable. Given a Banach space $X$, we show that if the kernel of a quotient map from some $\mathcal L_1$-space onto $X$ has the BAP then every kernel of every quotient map from any $\mathcal L_1$-space onto $X$ has the BAP. The dual result for $\mathcal L_\infty$-spaces also hold: if for some $\mathcal L_\infty$-space $E$ some quotient $E/X$ has the BAP then for every $\mathcal L_\infty$-space $E$ every quotient $E/X$ has the BAP.