The weakly ∞-compact approximation property and covering properties of weakly null sequences

Abstract

We introduce the property (\(\mathcal W_{\infty}\)) and the weakly \(\infty\)-compact approximation property (WICAP) of a Banach space \(X\). We establish a characterization of the property (\(\mathcal W_{\infty}\)) and relationships of the property (\(\mathcal W_{\infty}\)), the approximate identities for the algebra \(\mathcal W_{\infty}(X)\) and the WICAP. As a consequence, we obtain that both \(\ell_p (1 < p < \infty)\) and \(c_0\) fail property \((\mathcal{W}_{\infty})\). It is also shown that the WICAP is strictly stronger than the weakly compact approximation property.