Connective C⁎-algebras

Connectivity is a homotopy invariant property of separable C⁎-algebras which has three notable consequences: absence of nontrivial projections, quasidiagonality and a more geometric realization of KK-theory for nuclear C⁎-algebras using asymptotic morphisms. The purpose of this paper is to further explore the class of connective C⁎-algebras. We give new characterizations of connectivity for exact and for nuclear separable C⁎-algebras and show that an extension of connective separable nuclear C⁎-algebras is connective. We establish connectivity or lack of connectivity for C⁎-algebras associated to certain classes of groups: virtually abelian groups, linear connected nilpotent Lie groups and linear connected semisimple Lie groups.