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

Fixed a Banach operator ideal AA, we introduce and investigate two new approximation properties, which are strictly weaker than the bounded approximation property (BAP) for AA of Lima et al. (2010). We call them the weak BAP for AA and the local BAP for AA, 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 AA with approximation properties given by tensor norms and show that the Saphar BAP of order pp is the weak BAP for the ideal of absolutely p∗p∗-summing operators, 1≤p≤∞1≤p≤∞, 1/p+1/p∗=11/p+1/p∗=1.