2012/08/03 by Helge Glockner, Glockner, Helge
Mathematics · #22E65 #FOS: Mathematics #Functional Analysis (math.FA) #math.FA #msc:22E65
paper · pdf · doi:10.48550/arxiv.1208.0715
68 pages; v5: update of references, added two useful lemmas (Lemma 7.8, Lemma 7.10)
arxiv created 2016/02/06 · arxiv updated 2016/02/09
Let G be a Lie group modelled on a locally convex space, with Lie algebra g, and k be a non-negative integer or infinity. We say that G is Ck-semiregular if each Ck-curve c in g admits a left evolution Evol(c) in G. If, moreover, the map taking c to evol(c):=Evol(c)(1) is smooth, then G is called Ck-regular. For G a Ck-semiregular Lie group and m an order of differentiability, we show that evol is Cm if and only if Evol is Cm. If evol is continuous at 0, then evol is continuous. If G is a C0-semiregular Lie group, then continuity of evol implies its smoothness (so that G will be C0-regular), if smooth homomorphisms from G to C0-regular Lie groups separate points on G and g is (e.g.) sequentially complete. Further criteria for regularity properties are provided, and used to prove regularity for several important classes of Lie groups. Notably, we find that the Lie group Diff(M) of smooth diffeomorphisms of a paracompact finite-dimensional smooth manifold M (which need not be sigma-compact) is C1-regular. We also provide tools which enable to show that the Lie group of analytic diffeomorphisms of a compact real analytic manifold is C1-regular.