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Session 17. Functional Analysis: relations to Complex Analysis and PDE
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Complexifications of infinite-dimensional manifolds and new constructions of
infnite-dimensional Lie groups
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Helge Glöckner;, Universität Paderborn;, Germany
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The talk is based on the joint work with R. Dahmen and A. Schmeding
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We prove existence and uniqueness results for complexifications
of infinite-dimensional real analytic manifolds.
For each Banach-Lie group \(H\) over the real or complex field \({\mathbb K}\)
and each non-empty compact subset \(K\) of a regular \({\mathbb K}\)-analytic
manifold \(M\) modeled on a metrizable locally convex space, this enables
us to turn the group \(\mbox{Germ}(K,H)\) of all germs of \(H\)-valued
\({\mathbb K}\)-analytic maps on open neighbourhoods of \(K\)
into a \(C^0\)-regular \({\mathbb K}\)-analytic Lie group.
In particular, \(C^\omega(M,H)\) is a \(C^0\)-regular real analytic Lie group
for each compact real analytic manifold \(M\).
Combining the results with a recent idea of Neeb and Wagemann,
we also obtain a \(C^0\)-regular real analytic Lie group structure
on \(C^\omega({\mathbb R},H)\).
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