Equivariant $KK$-theory and model categories

TL;DR

This paper develops a model category framework for equivariant KK-theory, establishing a stable model structure on categories of G-C*-algebras and connecting it with existing KK-theory models.

Contribution

It introduces a new stable model structure for equivariant KK-theory, generalizing previous non-equivariant models and correcting earlier errors.

Findings

Constructed a stable model structure on G-C*-algebras

Connected the model category to the stable ∞-category KK^G_{sep}

Fixed critical errors in prior work by Joachim-Johnson

Abstract

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.