  For a completely Hausdorff quasi-topological group $G$, we construct a
universal pro-$C^*$-algebra $C(E^+G)$ as the non-commutative geometer's
analogue of the total space $EG$ of the classifying principal $G$-bundle $EG\to
BG$. The pro-$C^*$-algebra $C(EG)$ of (possibly unbounded) continuous functions
on $EG$ is then recoverable as the abelianization of $C(E^+G)$. Along the way,
we develop various aspects of the theory of quasi-topological $G$-spaces and
$G$-pro-$C^*$-algebras.
