Lie Group Structures on Quotient Groups and Universal Complexifications for Infinite-Dimensional Lie Groups

We characterize the existence of Lie group structures on quotient groups and the existence of universal complexifications for the class of Baker–Campbell–Hausdorff (BCH–) Lie groups, which subsumes all Banach–Lie groups and “linear” direct limit Lie groups, as well as the mapping groups CKr(M,G)≔{γ∈Cr(M,G) : γ∣M\K=1}, for every BCH–Lie group G, second countable finite-dimensional smooth manifold M, compact subset K of M, and 0⩽r⩽∞. Also the corresponding test function groups Dr(M,G)=⋃KCKr(M,G) are BCH–Lie groups.