论文标题
Banach Lie群体中的亚组接近
Subgroup proximity in Banach Lie groups
论文作者
论文摘要
让$ u $为Banach Lie Group，而$ G \ le U $紧凑型亚组。我们表明，在足够小的社区中包含的$ u $的封闭子组$ v \ supseteq g $是紧凑的，并通过接近$ 1 \ in u $ $ 1 \;这概括了蒙哥马利和Zippin的众所周知的结果。在此过程中，我们还证明了约旦关于一般线性群体有限亚组的定理：在足够小的社区中包含的$ U $的有限亚组$ v \ supseteq g $具有正常的Abelian Abelian Index索引子组，该索引是$ g \ le u u $ $ g \ le u $。 此外，$ u $的各种紧凑型亚组的空间，配备了$ u $的完整度量的Hausdorff公制，被证明是分析性的Banach歧管； （a）给定，固定尺寸或（b）紧凑型（可能是断开的）半密布亚组的（a）紧凑组的情况。最后，我们还证明，在适当的意义上，$ u $的紧凑型亚组的中心尺（或归一化）的操作是连续的（分别是上半连接的）。
Let $U$ be a Banach Lie group and $G\le U$ a compact subgroup. We show that closed Lie subgroups of $U$ contained in sufficiently small neighborhoods $V\supseteq G$ are compact, and conjugate to subgroups of $G$ by elements close to $1\in U$; this generalizes a well-known result of Montgomery and Zippin's from finite- to infinite-dimensional Lie groups. Along the way, we also prove an approximate counterpart to Jordan's theorem on finite subgroups of general linear groups: finite subgroups of $U$ contained in sufficiently small neighborhoods $V\supseteq G$ have normal abelian subgroups of index bounded in terms of $G\le U$ alone. Additionally, various spaces of compact subgroups of $U$, equipped with the Hausdorff metric attached to a complete metric on $U$, are shown to be analytic Banach manifolds; this is the case for both (a) compact groups of a given, fixed dimension, or (b) compact (possibly disconnected) semisimple subgroups. Finally, we also prove that the operation of taking the centralizer (or normalizer) of a compact subgroup of $U$ is continuous (respectively upper semicontinuous) in the appropriate sense.
