Non-Commutative classifying spaces of groups via quasi-topologies and pro-$C^{\ast }$-algebras

For a completely Hausdorff quasi-topological group $G$, we construct a universal pro-$C^{\ast }$-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-$C^{\ast }$-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-$C^{\ast }$-algebras.