Subgroup Proximity in Banach Lie Groups

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.