Fixed points for bounded orbits in Hilbert spaces
Fixed points for bounded orbits in Hilbert spaces
复制标题
希尔伯特空间中有界轨道的不动点
DOI：
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
N. Monod
中科院分区：
文献类型：
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作者：
Maxime Gheysens;N. Monod
考虑拓扑群G的以下性质：Hilbert空间上具有有界轨道的每个连续仿射G-作用都有不动点。我们证明了这一性质的顺从性局部紧σ紧群（例如可数群）的特点。
沿着的方式，我们引入了一个“温和”的变种的经典归纳表示和我们推广的Gaboriau-里昂定理证明，任何不服从局部紧群承认离散自由子群的概率变体。这导致了“测度理论的解决方案”的冯诺依曼问题的局部紧群。
我们说明后一个结果给出了局部紧群的Dixandom问题的部分答案。
Consider the following property of a topological group G: every continuous affine G-action on a Hilbert space with a bounded orbit has a fixed point. We prove that this property characterizes amenability for locally compact sigma-compact groups (e.g. countable groups).
Along the way, we introduce a "moderate" variant of the classical induction of representations and we generalize the Gaboriau--Lyons theorem to prove that any non-amenable locally compact group admits a probabilistic variant of discrete free subgroups. This leads to the "measure-theoretic solution" to the von Neumann problem for locally compact groups.
We illustrate the latter result by giving a partial answer to the Dixmier problem for locally compact groups.