Weak amenability of the second dual of a Banach algebra
Volume 182 / 2007
Abstract
It is known that a Banach algebra $\mathcal A$ inherits amenability from its second Banach dual ${\mathcal A}^{**}$. No example is yet known whether this fails if one considers the weak amenability instead, but the property is known to hold for the group algebra $L^1(G)$, the Fourier algebra $A(G)$ when $G$ is amenable, the Banach algebras $\mathcal A$ which are left ideals in $\mathcal A^{**}$, the dual Banach algebras, and the Banach algebras $\mathcal A$ which are Arens regular and have every derivation from $\mathcal A$ into $\mathcal A^*$ weakly compact. In this paper, we extend this class of algebras to the Banach algebras for which the second adjoint of each derivation $D:\mathcal A\to \mathcal A^{*}$ satisfies $D''(\mathcal A^{**})\subseteq \mathop{\rm WAP}\nolimits(\mathcal A)$, the Banach algebras $\mathcal A$ which are right ideals in $\mathcal A^{**}$ and satisfy $\mathcal A^{**}\mathcal A=\mathcal A^{**}$, and to the Figà-Talamanca–Herz algebra $A_p(G)$ for $G$ amenable. We also provide a short proof of the interesting recent criterion on when the second adjoint of a derivation is again a derivation.
