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

摘要

Fixed a Banach operator ideal $\mathcal A$, we introduce and investigate twonew approximation properties, which are strictly weaker than the boundedapproximation property (BAP) for $\mathcal A$ of Lima, Lima and Oja (2010). Wecall 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 thisframework, we address the question of approximation properties passing fromdual spaces to underlying spaces. We relate the weak and local BAPs for$\mathcal A$ with approximation properties given by tensor norms and show thatthe Saphar BAP of order $p$ is the weak BAP for the ideal of absolutely$p^*$-summing operators, $1\leq p\leq\infty$, $1/p + 1/{p^*}=1$.