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.