Second cohomology group of group algebras with coefficients in iterated duals

In this paper we show that the first cohomology group $\mathcal{H}^1(\ell^1(G),(\ell^1(S))^{(n)})$ is zero for every odd $n\in\mathbb{N}$ and for every $G$-set $S$. In the case when $G$ is a discrete group, this is a generalization of the following result of Dales et al.: for any locally compact group $G$, $L^1(G)$ is $(2n+1)$-weakly amenable.

Next we show that the second cohomology group $\mathcal{H}^2(\ell^1(G),(\ell^1(S))^{(n)})$ is a Banach space. Finally, for every locally compact group $G$ we show that $\mathcal{H}^2(L^1(G),(L^1(G))^{(n)})$ is a Banach space for every odd $n\in\mathbb{N}$.