We initiate a study of maximal subgroups and maximal von Neumann subalgebras which have the Haagerup property. We determine maximal Haagerup subgroups inside $\mathbb{Z}^2 \rtimes \operatorname{SL}_2(\mathbb{Z})$ and obtain several explicit instances where maximal Haagerup subgroups yield maximal Haagerup subalgebras. Our techniques are on one hand based on group-theoretic considerations, and on the other on certain results on intermediate von Neumann algebras, in particular these allowing us to deduce that all the intermediate algebras for certain inclusions arise from groups or from group actions. Some remarks and examples concerning maximal non-(T) subgroups and subalgebras are also presented, and we answer two questions of Ge regarding maximal von Neumann subalgebras.

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具有 Haagerup 性质的极大子群和冯诺依曼子代数

我们开始研究具有 Haagerup 性质的极大子群和极大冯诺依曼子代数。我们确定 $\mathbb{Z}^2 \rtimes \operatorname{SL}_2(\mathbb{Z})$ 内的最大 Haagerup 子群，并获得几个显式实例，其中最大 Haagerup 子群产生最大 Haagerup 子代数。我们的技术一方面基于群论考虑，另一方面基于中间冯诺依曼代数的某些结果，特别是这些结果使我们能够推断出某些包含的所有中间代数来自群或群作用。还给出了一些关于极大非（T）子群和子代数的评论和例子，我们回答了 Ge 关于极大冯诺依曼子代数的两个问题。