非交换拓扑边界和可变随机中间子代数

Shuoxing Zhou
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引用次数: 0

摘要

作为离散群 $\Gamma$ 的拓扑边界的类比,我们定义了三维冯-诺依曼代数$(M,\tau)$ 的非交换拓扑边界,并将其应用于归纳 [AHO23] 的主要结果,表明对于可三维冯-诺依曼代数上的迹保留作用 $\Gamma \curvearrowright(A. \tau_A)\tau_A)$ 时,$\Gamma$不变度量$mu\inmathrm{Prob}(\mathrm{SA}(\Gamma\ltimes A))$ 必须支持在 $A$ 和 $\Gamma\ltimes A$ 之间的可变中间子代数上。通过对自由 p.m.p. 作用 $\Gamma\curvearrowright(X,\nu_X)$ 取$(A,\tau)=L^\infty(X,\nu_X)$,我们得到了 $\mathcal{R}_{Gamma\curvearrowright X}$ 的无变量随机子等价关系的类似结果。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Noncommutative topological boundaries and amenable invariant random intermediate subalgebras
As an analogue of topological boundary of discrete groups $\Gamma$, we define the noncommutative topological boundary of tracial von Neumann algebras $(M,\tau)$ and apply it to generalize the main results of [AHO23], showing that for a trace preserving action $\Gamma \curvearrowright(A,\tau_A)$ on an amenable tracial von Neumann algebra, a $\Gamma$-invariant measure $\mu\in\mathrm{Prob}(\mathrm{SA}(\Gamma\ltimes A))$ supported on amenable intermediate subalgebras between $A$ and $\Gamma\ltimes A$ is necessary supported on the subalgebras of $\mathrm{Rad}(\Gamma)\ltimes A$. By taking $(A,\tau)=L^\infty(X,\nu_X)$ for a free p.m.p. action $\Gamma \curvearrowright(X,\nu_X)$, we obtain a similar results for the invariant random subequivalence relations of $\mathcal{R}_{\Gamma \curvearrowright X}$.
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