Analytic mappings between LB-spaces and applications in infinite-dimensional Lie theory

Abstract

We give a sufficient criterion for complex analyticity of nonlinear maps defined on direct limits of normed spaces. This tool is then used to construct new classes of (real and complex) infinite dimensional Lie groups: The group DiffGerm(K , X ) of germs of analytic diffeomorphisms around a compact set K in a Banach space X and the group n∈N G n where the G n are Banach Lie groups. IntroductionAn infinite dimensional analytic Lie group is a group which is at the same time an analytic manifold modeled on some locally convex topological vector space such that the group operations are analytic. To construct such Lie groups, it is useful to have tools at hand ensuring the analyticity of nonlinear mappings between locally convex spaces. This text provides a sufficient criterion for complex analyticity in the case where the domain is an LB-space, i.e. a locally convex direct limit of an ascending sequence of Banach spaces:Theorem A [Complex analytic mappings defined on LB-spaces] Let E be a C-vector space that is the union of the increasing sequence of subspaces (E n ) n∈N . Assume that a norm · E n is given on each E n such that all bonding maps i n : E n −→ E n+1 : x → x are continuous and have an operator norm at most 1. We give E the locally convex direct limit topology and assume that it is Hausdorff. Let R > 0 and let U := n∈N B E n R (0) be the union of all open balls with radius R around 0. Let f : U −→ F be a function defined on U with values in a locally convex space F, such that each f n := f | B En R (0) is C-analytic and bounded. Then f is C-analytic. R. Dahmen (B)

Completeness of Infinite-dimensional Lie Groups in Their Left Uniformity

“…Beyond unit groups of Banach algebras, let us consider an ascending sequence G 1 ⊆ G 2 ⊆ · · · of Banach-Lie groups over K ∈ {R, C} such that each inclusion map G n → G n+1 is a Kanalytic group homomorphism. In [13,Theorem C], conditions were spelled out which ensure that G = n∈N G n can be made a K-analytic Lie group modelled on the locally convex direct limit of the respective Lie algebras. We show that product sets are large in the Lie groups G constructed in loc.…”

“…Since R ′ (x, y) op ≤ 1 2 for all (x, y) ∈ B g s 0 (0) × B g s 0 (0) by (13) and the latter set is convex, 1.2 (b) shows that Lip(R) ≤ 1 2 . For x ∈ B g s 0 (0), consider the map µ g x : B g s 0 (0) → g, y → µ g (x, y).…”

The tame Butcher group

“…In the last chapter we construct Lie group morphisms from the tame Butcher group to a class of Lie groups of germs of diffeomorphisms. Consider a real Banach space E and recall from [Dah10] that the group DiffGerm({(y 0 , 0)}, E × R) of germs of real analytic diffeomorphisms which fix the point (y 0 , 0) in E × R is a Lie group. For finite-dimensional E, the construction goes back to [Glö07] (compare also [KR01] and [Lei94]).…”

“…y 0 we deduce from the proof of loc.cit. (see [Dah10,p.118]) for all y ∈ B E R 2 (y 0 ) the estimates…”

“…the locally convex inductive limit of Banach spaces. In fact Germ({y 0 }, E) {y0} is again a Hausdorff space by [Dah10,Proposition 3.3. ].…”