标题： 与乘积群的$W^*$-动力系统相关的非交换中间因子定理 显示英文标题

标题： Non-commutative Intermediate Factor theorem associated with $W^*$-dynamics of product groups

摘要： 设$G = G_{1} \times G_{2}$为两个局部紧致、第二可数群的乘积，且$\mu \in \mathrm{Prob}(G)$为$\mu = \mu_{1} \times \mu_{2}$的形式，其中$\mu_{i} \in \mathrm{Prob}(G_{i})$。 设$(B,\nu_B)$为相关的泊松边界。 我们证明每个中间 $G$-冯诺依曼代数 $\mathcal{M}$，具有 \[ \mathcal{N} \subseteq \mathcal{M} \subseteq \mathcal{N} \,\bar{\otimes}\, L^{\infty}(B,\nu) \]，可以分解为形式 $\mathcal{N}\bar{\otimes}L^{\infty}(C,\nu_C)$，其中 $(C,\nu_C)$是一个 $(G,\mu)$-边界。 此处，$\mathcal{N}$是一个冯诺依曼代数，$G$在其上以保持迹的方式作用。这推广了Bader--Shalom（\cite[Theorem~1.9]{BS06}）在可测设置下证明的中间因子定理。此外，我们给出了与$W^{*}$-动力系统相关的分裂现象的各种其他例子。我们还表明，某些假设对于中间代数的分裂是必要的，环境张量积代数中的理想会阻碍分裂现象。我们还使用来自\cite{glasner2023intermediate}的主定理来肯定地解决\cite[Problem~5.2]{jiangskalski}的第二部分。 显示英文摘要

摘要： Let $G = G_{1} \times G_{2}$ be a product of two locally compact, second countable groups and $\mu \in \mathrm{Prob}(G)$ be of the form $\mu = \mu_{1} \times \mu_{2}$, where $\mu_{i} \in \mathrm{Prob}(G_{i})$. Let $(B,\nu_B)$ be the associated Poisson boundary. We show that every intermediate $G$-von Neumann algebra $\mathcal{M}$ with \[ \mathcal{N} \subseteq \mathcal{M} \subseteq \mathcal{N} \,\bar{\otimes}\, L^{\infty}(B,\nu) \] splits as a tensor product of the form $\mathcal{N}\bar{\otimes}L^{\infty}(C,\nu_C)$, where $(C,\nu_C)$ is a $(G,\mu)$-boundary. Here, $\mathcal{N}$ is a tracial von Neumann algebra on which $G$ acts trace-preservingly. This generalizes the Intermediate Factor Theorem proved by Bader--Shalom (\cite[Theorem~1.9]{BS06}) in the measurable setup. In addition, we give various other examples of the splitting phenomenon associated with $W^{*}$-dynamics. We also show that certain assumptions are necessary for the intermediate algebras to split, and ideals in the ambient tensor product algebra obstruct the splitting phenomenon. We also use the Master theorem from \cite{glasner2023intermediate} to resolve the second part of \cite[Problem~5.2]{jiangskalski} in the affirmative.