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Abstract view

Lie Groups of Measurable Mappings

We describe new construction principles for infinite-dimensional Lie groups. In particular, given any measure space $(X,\Sigma,\mu)$ and (possibly infinite-dimensional) Lie group $G$, we construct a Lie group $L^\infty (X,G)$, which is a Fr\'echet-Lie group if $G$ is so. We also show that the weak direct product $\prod^*_{i\in I} G_i$ of an arbitrary family $(G_i)_{i\in I}$ of Lie groups can be made a Lie group, modelled on the locally convex direct sum $\bigoplus_{i\in I} L(G_i)$.
 MSC Classifications: 22E65 - Infinite-dimensional Lie groups and their Lie algebras: general properties [See also 17B65, 58B25, 58H05] 46E40 - Spaces of vector- and operator-valued functions 46E30 - Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Kothe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 22E67 - Loop groups and related constructions, group-theoretic treatment [See also 58D05] 46T20 - Continuous and differentiable maps [See also 46G05] 46T25 - Holomorphic maps [See also 46G20]