On algebras associated with invariant means on the subnormal subgroups of an amenable group

Let $G$ be an amenable group. We define and study an algebra $\mathcal{A}_{sn}(G)$, which is related to invariant means on the subnormal subgroups of $G$. For a just infinite amenable group $G$, we show that $\mathcal{A}_{sn}(G)$ is nilpotent if and only if $G$ is not a branch group, and in the case that it is nilpotent we determine the index of nilpotence. We next study $\operatorname{rad} \ell^1(G)^{**}$ for an amenable branch group $G$, and show that it always contains nilpotent left ideals of arbitrarily large index, as well as non-nilpotent elements. This provides infinitely many finitely-generated counterexamples to a question of Dales and Lau, first resolved by the author in a previous article, which asks whether we always have $(\operatorname{rad} \ell^1(G)^{**})^{\Box 2} = \{ 0 \}$. We further study this question by showing that $(\operatorname{rad} \ell^1(G)^{**})^{\Box 2} = \{ 0 \}$ imposes certain structural constraints on the group $G$.