张量积的分割和中间因子定理:连续版本

Tattwamasi Amrutam, Yongle Jiang
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引用次数: 0

摘要

让 $G$ 是一个离散群。给定单元$G$-$C^*$-代数$mathcal{A}$和$mathcal{B}$,我们给出一个抽象条件,在这个条件下,形式为$mathcal{A}/subset \mathcal{C}/subset\mathcal{A}/otimes_{text{min}}/mathcal{B}$的每个$G$-子代数$mathcal{C}$都是张量积。这概括了扎哈里亚斯(Zacharias)和兹西多(Zsido)在$C^*$-代数中著名的分裂结果。作为应用,我们证明了中间因子定理的拓扑版本。当一个乘积组$G=\Gamma_1\times\Gamma_2$(通过乘积作用)作用于对应的$\Gamma_i$边界$\partial\Gamma_i$的乘积时,使用抽象条件、我们证明每个中间子代数$C(X)/subset/mathcal{C}/subset C(X)\otimes_{text{min}}C(\partial\Gamma_1\times\partial\Gamma_2)$ 是一个张量积(在关于$X$的一些附加假设下)。这可以看作是中间因子定理的拓扑版本。我们将证明我们的假设是必要的,一般情况下不能放松。我们还为 $C^*$-gebras 引入了均匀刚性作用的概念,并用它给出了每一个不变的中间代数都是张量乘的各类夹杂$/mathcal{A}/subset \mathcal{A}/otimes_{/text{min}}/mathcal{B}$。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Splitting of Tensor Products and Intermediate Factor Theorem: Continuous Version
Let $G$ be a discrete group. Given unital $G$-$C^*$-algebras $\mathcal{A}$ and $\mathcal{B}$, we give an abstract condition under which every $G$-subalgebra $\mathcal{C}$ of the form $\mathcal{A}\subset \mathcal{C}\subset \mathcal{A}\otimes_{\text{min}}\mathcal{B}$ is a tensor product. This generalizes the well-known splitting results in the context of $C^*$-algebras by Zacharias and Zsido. As an application, we prove a topological version of the Intermediate Factor theorem. When a product group $G=\Gamma_1\times\Gamma_2$ acts (by a product action) on the product of corresponding $\Gamma_i$-boundaries $\partial\Gamma_i$, using the abstract condition, we show that every intermediate subalgebra $C(X)\subset\mathcal{C}\subset C(X)\otimes_{\text{min}}C(\partial\Gamma_1\times \partial\Gamma_2)$ is a tensor product (under some additional assumptions on $X$). This can be considered as a topological version of the Intermediate Factor theorem. We prove that our assumptions are necessary and cannot generally be relaxed. We also introduce the notion of a uniformly rigid action for $C^*$-algebras and use it to give various classes of inclusions $\mathcal{A}\subset \mathcal{A}\otimes_{\text{min}}\mathcal{B}$ for which every invariant intermediate algebra is a tensor product.
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