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Lie group extensions associated to projective modules of continuous inverse algebras. (English) Zbl 1212.22009
Summary: 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 $$\mathrm {GL}_A(E)$$ of automorphisms of the $$A$$-module $$E$$. This Lie group extension is a “non-commutative” version of the group $$\mathrm{Aut}({\mathbb V})$$ of automorphisms of a vector bundle over a compact manifold $$M$$, which arises for $$G = \mathrm{Diff}(M)$$, $$A = C^{\infty }(M,{\mathbb C})$$ 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 $$E$$.

##### MSC:
 22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties 58B34 Noncommutative geometry (à la Connes)
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