Connective C*-algebras

TL;DR

This paper investigates the properties of connective C*-algebras, providing new characterizations and exploring their behavior in various classes of group-related C*-algebras, highlighting their geometric and algebraic significance.

Contribution

It offers new characterizations of connectivity in exact and nuclear C*-algebras and analyzes connectivity in C*-algebras associated with specific classes of groups.

Findings

Connectivity is preserved under extensions of nuclear C*-algebras.

Certain group C*-algebras are shown to be connective or not, depending on the group class.

Connectivity relates to geometric and algebraic properties of C*-algebras.

Abstract

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.