Lie group extensions associated to projective modules of continuous inverse algebras

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