A More General Method to Classify up to Equivariant $KK$-Equivalence

Abstract

Using a homological invariant together with an obstruction class in a certain Ext${}^2$-group, we classify objects in triangulated categories that have projective resolutions of length two. This invariant gives strong classification results for actions of the circle group on $C^{\ast }$-algebras, $C^{\ast }$-algebras over finite unique path spaces, and graph $C^{\ast }$-algebras with finitely many ideals.