Weak amenability for the second dual of Banach modules (Q317733)

The authors modify the notion of weak \(\Delta\)-amenability given in [\textit{D. Ebrahimi Bagha} and \textit{M. Amini}, Cubo 13, No. 2, 127--137 (2011; Zbl 1258.46020)] by considering the closure of a certain space. For a Banach algebra \(A\), a Banach \(A\)-module \(E\) and a bounded Banach \(A\)-bimodule homomorphism \(\Delta: E\to A\), they present some sufficient conditions implying the weak \(\Delta\)-amenability of \(E\). They show that the weak \(\Delta''\)-amenability of \(E''\), the second dual of \(E\), ensures the weak \(\Delta\)-amenability of \(E\). These results can be regarded as the generalizations of some known results in the literature; see [\textit{A. Bodaghi} et al., Banach J. Math. Anal. 3, No. 1, 131--142 (2009; Zbl 1163.46034)].