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

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 $p^{\ast }$-summing operators, $1\le p\le \mathrm{\infty }$, $1\mathrm{/}p+1\mathrm{/}{p}^{\ast }=1$.