Weaker relatives of the bounded approximation property for a Banach operator ideal

TL;DR

This paper introduces and analyzes two new approximation properties for Banach operator ideals, called the weak BAP and local BAP, which are weaker than the classical bounded approximation property, and explores their relationships and implications.

Contribution

It defines the weak and local BAPs for Banach operator ideals, establishing their hierarchy, and relates these properties to tensor norms and known approximation properties.

Findings

The local BAP is strictly weaker than the weak BAP.

The weak BAP for the ideal of absolutely p-summing operators coincides with the Saphar BAP of order p.

Approximation properties can pass from dual spaces to original spaces under certain conditions.

Abstract

Fixed a Banach operator ideal $\mathcal A$, we introduce and investigate two new approximation properties, which are strictly weaker than the bounded approximation property (BAP) for $\mathcal A$ of Lima, Lima and Oja (2010). We call them the weak BAP for $\mathcal A$ and the local BAP for $\mathcal A$, showing that the latter is in turn strictly weaker than the former. Under this framework, we address the question of approximation properties passing from dual spaces to underlying spaces. We relate the weak and local BAPs for $\mathcal A$ with approximation properties given by tensor norms and show that the Saphar BAP of order $p$ is the weak BAP for the ideal of absolutely $p^*$-summing operators, $1\leq p\leq\infty$, $1/p + 1/{p^*}=1$.