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 $\textrm{rad}\, \ell^1(G)^{**}$ 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 [4], first resolved by the author in [10], which asks whether we always have $(\textrm{rad}\, \ell^1(G)^{**})^{\Box 2} = \{0 \}$. We further study this question by showing that $(\textrm{rad}\, \ell^1(G)^{**})^{\Box 2} = \{0 \}$ imposes certain structural constraints on the group G.