Equivariant $KK$-theory and model categories

We cast Kasparov's equivariant KK-theory in the framework of model categories. We obtain a stable model structure on a certain category of locally multiplicative convex $G$-$C^*$-algebras, which naturally contains the stable $\infty$-category $KK^G_{\operatorname{sep}}$ as described by Bunke, Engel, Land (\cite{BEL}). Non-equivariantly, $KK$-theory was studied using model categories by Joachim-Johnson (\cite{MJ}). We generalize their ideas in the equivariant case, and also fix some critical errors that their work had.