Direct limits of regular Lie groups

Abstract

Let G be a regular Lie group which is a directed union of regular Lie groups (all modelled on possibly infinite‐dimensional, locally convex spaces). We show that as a regular Lie group if G admits a so‐called direct limit chart. Notably, this allows the regular Lie group of compactly supported diffeomorphisms to be interpreted as a direct limit of the regular Lie groups of diffeomorphisms supported in compact sets , even if the finite‐dimensional smooth manifold M is merely paracompact (but not necessarily σ‐compact), which is new. Similar results are obtained for the test function groups with values in a Lie group F.

Mapping groups associated with real-valued function spaces and direct limits of Sobolev-Lie groups

“…Example 1.3 For m ∈ N, k ∈ N 0 , and α ∈ ]0, 1], Proposition 1.1 can be applied to the Banach spaces F (U, R) := C k,α (U, R) of k times Hölderdifferentiable functions on bounded open subsets U ⊆ R m (see [18]). This yields Banach-Lie groups C k,α (M, G) := F (M, G) with properties as described in the proposition.…”

“…See [36] for a characterization. Example 1.4 For m = 1 and p ∈ [1, ∞], Proposition 1.1 can be applied to the Banach spaces F (U, R) := AC L p (U, R) of absolutely continuous functions with L p -derivatives on bounded open intervals U = ∅ in R (see [18]). For M := S the unit circle, this yields Banach-Lie groups AC L p (S, G) := F (S, G) with properties as described in the proposition.…”