We call a unital locally convex algebra $A$ a continuous inverse algebra if its unit group $A^\times $ is open and inversion is a continuous map. For any smooth action of a, possibly infinite-dimensional, connected Lie group $G$ on a continuous inverse algebra $A$ by automorphisms and any finitely generated projective right $A$-module $E$, we construct a Lie group extension $\widehat{G}$ of $G$ by the group $\operatorname{GL}_A(E)$ of automorphisms of the $A$-module $E$. This Lie group extension is a “non-commutative” version of the group $\operatorname{Aut}({\mathbb{V}})$ of automorphism of a vector bundle over a compact manifold $M$, which arises for $G = \operatorname{Diff}(M)$, $A = C^\infty (M,{\mathbb{C}})$ and $E = \Gamma {\mathbb{V}}$. We also identify the Lie algebra $\widehat{\mathfrak{g}}$ of $\widehat{G}$ and explain how it is related to connections of the $A$-module $E$.
@article{127115,
author = {Karl-Hermann Neeb},
title = {Lie group extensions associated to projective modules of continuous inverse algebras},
journal = {Archivum Mathematicum},
volume = {044},
year = {2008},
pages = {465-489},
zbl = {1212.22009},
mrnumber = {2501579},
language = {en},
url = {http://dml.mathdoc.fr/item/127115}
}
Neeb, Karl-Hermann. Lie group extensions associated to projective modules of continuous inverse algebras. Archivum Mathematicum, Tome 044 (2008) pp. 465-489. http://gdmltest.u-ga.fr/item/127115/
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