Non-stable K-theory and extremally rich C*-algebras

TL;DR

This paper investigates key K-theoretic properties of extremally rich C*-algebras, establishing several properties for specific subclasses and exploring their interrelations and stability under extensions.

Contribution

It proves that extremally rich C*-algebras with weak cancellation possess multiple K-theoretic properties and that these properties are preserved under extensions.

Findings

All four properties hold for isometrically rich C*-algebras.

Extremally rich C*-algebras that are purely infinite or of real rank zero satisfy these properties.

Weak cancellation implies the other properties for extremally rich C*-algebras.

Abstract

We consider the properties weak cancellation, K_1-surjectivity, good index theory, and K_1-injectivity for the class of extremally rich C*-algebras, and for the smaller class of isometrically rich C*-algebras. We establish all four properties for isometrically rich C*-algebras and for extremally rich C*-algebras that are either purely infinite or of real rank zero, K_1-injectivity in the real rank zero case following from a prior result of H. Lin. We also show that weak cancellation implies the other properties for extremally rich C*-algebras and that the class of extremally rich C*-algebras with weak cancellation is closed under extensions. Moreover, we consider analogous properties which replace the group K_1(A) with the extremal K-set K_e(A) as well as two versions of K_0-surjectivity.