来自克隆系统的类胰蛋白酶基团的Von Neumann algebras
Von Neumann algebras of Thompson-like groups from cloning systems
摘要：
我们证明了由所谓的$d$ -ary克隆系统所产生的类Guppson群所对应的群von Neumann代数的各种结果。克隆系统是Witzel和第二作者开发的一个框架，随后由Skipper和第二作者开发了一个$d$ -ary版本，它可以用于构建经典Thompson群$F$、$T$和$V$的推广。给定一族具有$d$ -元克隆系统的群$(G_n)_{n\in\mathbb{N}}$，我们得到了一个类PIMSON群$\mathscr{T}_d(G_*)$，并给出了群von Neumann代数$\mathcal{L}(\mathscr{T}_d(G_*))$具有理想性质的一些温和的自然条件.例如，如果$d$ -ary克隆系统是“完全相容的”和“多样的”，那么我们证明$\mathcal{L}(\mathscr{T}_d(G_*))$是一个$\textrm{II}_1$型因子。如果$d$多元克隆系统是“均匀的”和“稍纯的”，则证明$\mathcal{L}(\mathscr{T}_d(G_*))$甚至是McDuff因子，因此$\mathscr{T}_d(G_*)$是内在服从的.满足这些条件的$d$ -ary克隆系统的例子很容易得到，包括许多现有的例子，例如我们的结果表明，对于$bV$和$bF$，Brin-Dehornoy辫子Thompson群和纯辫子Thompson群，$\mathcal{L}(bV)$和$\mathcal{L}(bF)$是$\textrm{II}_1$型因子，$\mathcal{L}(bF)$是McDuff。特别地，我们得到了令人惊讶的结果，即$bF$是内在顺应的。
英文摘要：
We prove a variety of results about the group von Neumann algebras associated to Thompson-like groups arising from so called $d$-ary cloning systems. Cloning systems are a framework developed by Witzel and the second author, with a $d$-ary version subsequently developed by Skipper and the second author, which can be used to construct generalizations of the classical Thompson's groups $F$, $T$, and $V$. Given a family of groups $(G_n)_{n\in\mathbb{N}}$ with a $d$-ary cloning system, we get a Thompson-like group $\mathscr{T}_d(G_*)$, and in this paper we find some mild, natural conditions under which the group von Neumann algebra $\mathcal{L}(\mathscr{T}_d(G_*))$ has desirable properties. For instance, if the $d$-ary cloning system is "fully compatible" and "diverse" then we prove that $\mathcal{L}(\mathscr{T}_d(G_*))$ is a type $\textrm{II}_1$ factor. If moreover the $d$-ary cloning system is "uniform" and "slightly pure" then we prove $\mathcal{L}(\mathscr{T}_d(G_*))$ is even a McDuff factor, so $\mathscr{T}_d(G_*)$ is inner amenable. Examples of $d$-ary cloning systems satisfying these conditions are easy to come by, and include many existing examples, for instance our results show that for $bV$ and $bF$ the Brin-Dehornoy braided Thompson group and pure braided Thompson group, $\mathcal{L}(bV)$ and $\mathcal{L}(bF)$ are type $\textrm{II}_1$ factors and $\mathcal{L}(bF)$ is McDuff. In particular we get the surprising result that $bF$ is inner amenable.
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