论文标题
在与不变平均值相关的代数上
On algebras associated with invariant means on the subnormal subgroups of an amenable group
论文作者
论文摘要
让$ g $成为一个正式的组。我们定义和研究一个代数$ \ MATHCAL {a} _ {sn}（g）$，这与$ g $的亚正常亚组中的不变均值有关。对于仅仅是无限的umenable $ g $，我们表明$ \ mathcal {a} _ {sn}（g）$在且仅当$ g $不是分支机构时才nilpotent，并且如果它是nilpotent，我们确定nilpotence的索引。我们下一个研究$ \ operatorName {rad} \ ell^1（g）^{**} $对于不友善的分支组$ g $，并表明它始终包含任意较大索引的nilpotent左左左左索引，以及非尼尔特元素。这为作者在上一篇文章中首先解决了许多有限生成的反例，该问题最初是在上一篇文章中解决的，它询问我们是否总是有$（\ peripatorName {rad} \ ell^1（g）^{** {**} {**} {**}）我们通过表明$（\ operatorName {rad} \ ell^1（g）^{**}）^{\ box 2} = \ {0 \} $对$（\ ell^1（g）^{**}）$进一步研究了这个问题。
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 $G$.
