  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 $E$.

Operator Algebras;Rings and Algebras