Lie group extensions associated to projective modules of continuous inverse algebras

Abstract

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 $\hat G$ of $G$ by the group $\GL_A(E)$ of automorphisms of the $A$-module $E$. This Lie group extension is a ``non-commutative'' version of the group $\Aut(\V)$ of automorphism of a vector bundle over a compact manifold $M$, which arises for $G = \Diff(M)$, $A = C^\infty(M,\C)$ and $E = \Gamma\V$. We also identify the Lie algebra $\hat\g$ of $\hat G$ and explain how it is related to connections of the $A$-module $E$.