arXiv 2212.062552022-12-12math.GRmath.FAmath.GNmath.MG

Banach李群中的子群邻近性

Subgroup proximity in Banach Lie groups

Alexandru Chirvasitu

备注：: 28 pages + references

摘要：

设$U$是Banach李群，$G\le U$是紧子群.我们证明了包含在足够小的邻域$V\supseteq G$中的$U$的闭李群是紧的，并且通过接近$1\in U$的元素与$G$的子群共轭;这推广了Montgomery和Zippin从有限维到无限维李群的一个著名结果。一路上，我们还证明了一个近似对应的约旦定理的有限子群一般线性群：有限子群$U$包含在足够小的邻域$V\supseteq G$有正常的阿贝尔子群的指数有界的$G\le U$单独。此外，各种空间的紧子群的$U$，配备了Hausdorff度量附加到一个完整的度量$U$，被证明是解析巴拿赫流形;这是两种情况（a）紧群的一个给定的，固定的维度，或（b）紧（可能不连通）半单群。最后，我们还证明了取$U$的紧子群的中心化子（或正规化子）的运算在适当的意义下是连续的（分别是上连续的）。

英文摘要：

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.

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