形变的极大刚性子代数与$L^{2}$ -上同调
Maximal rigid subalgebras of deformations and $L^{2}$-cohomology
摘要：
在过去的二十年中，Sorin Popa的突破性变形/刚性理论为冯诺依曼代数$M$产生了显著的刚性结果，它可以在一个更大的代数$\widetilde M \supseteq M$内通过一个作用$α: \mathbb{R} \to {\rm Aut}(\widetilde M)$变形，同时包含关于该变形的子代数$Q$ { \it rigid}，也就是说，使得$α_t \to {\rm id}$在$Q$的单位球上均匀地作为$t \to 0$。然而，它仍然不清楚如何利用不同的刚性子代数之间的相互作用，而不是在指定的相对位置。我们表明，事实上，任何扩散的子代数，这是刚性的混合s-可塑变形包含在一个子代数，这是唯一的最大的刚性。特别地，由任何族的刚性子代数所生成的代数，其漫交本身必须相对于该变形是刚性的。这个家族有两个成员的情况是这项工作的动机，例如，如果$G$是一个可数群与$β^{1}_{(2)}(G) > 0$，那么$L(G)$不能产生的两个属性$(T)$子代数扩散交叉;然而，结果是最惊人的家庭是无限的。
英文摘要：
In the past two decades, Sorin Popa's breakthrough deformation/rigidity theory has produced remarkable rigidity results for von Neumann algebras $M$ which can be deformed inside a larger algebra $\widetilde M \supseteq M$ by an action $α: \mathbb{R} \to {\rm Aut}(\widetilde M)$, while simultaneously containing subalgebras $Q$ {\it rigid} with respect to that deformation, that is, such that $α_t \to {\rm id}$ uniformly on the unit ball of $Q$ as $t \to 0$. However, it has remained unclear how to exploit the interplay between distinct rigid subalgebras not in specified relative position. We show that in fact, any diffuse subalgebra which is rigid with respect to a mixing s-malleable deformation is contained in a subalgebra which is uniquely maximal with respect to being rigid. In particular, the algebra generated by any family of rigid subalgebras that intersect diffusely must itself be rigid with respect to that deformation. The case where this family has two members was the motivation for this work, showing for example that if $G$ is a countable group with $β^{1}_{(2)}(G) > 0$, then $L(G)$ cannot be generated by two property $(T)$ subalgebras with diffuse intersection; however, the result is most striking when the family is infinite.
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