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

TL;DR

This paper investigates algebraic structures related to invariant means on subnormal subgroups of amenable groups, revealing conditions for nilpotency and counterexamples to a longstanding question about the radical of the bidual of group algebras.

Contribution

It introduces and analyzes the algebra _{sn}(G) associated with invariant means, characterizes its nilpotency in certain groups, and constructs counterexamples to a question about the radical of (G)^{**}.

Findings

_{sn}(G) is nilpotent iff G is not a branch group.

The radical (G)^{**} contains nilpotent ideals of arbitrarily large index.

Counterexamples show ((G)^{**})^{\u22a2 2} can be non-zero, answering a question of Dales and Lau.

Abstract

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 $G$.