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Let $G$ be an amenable group. We define and study an algebra$\mathcal{A}_{sn}(G)$, which is related to invariant means on the subnormalsubgroups 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. Wenext study $\operatorname{rad} \ell^1(G)^{**}$ for an amenable branch group$G$, and show that it always contains nilpotent left ideals of arbitrarilylarge index, as well as non-nilpotent elements. This provides infinitely manyfinitely-generated counterexamples to a question of Dales and Lau, firstresolved by the author in a previous article, which asks whether we always have$(\operatorname{rad} \ell^1(G)^{**})^{\Box 2} = \{ 0 \}$. We further study thisquestion by showing that $(\operatorname{rad} \ell^1(G)^{**})^{\Box 2} = \{ 0\}$ imposes certain structural constraints on the group $G$.

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