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

We consider the properties of weak cancellation, K1K1-surjectivity, good index theory, and K1K1-injectivity, for the class of extremally rich C⁎C⁎-algebras, and for the smaller class of isometrically rich C⁎C⁎-algebras. We establish all four properties for isometrically rich C⁎C⁎-algebras and for extremally rich C⁎C⁎-algebras that are either purely infinite or of real rank zero, K1K1-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⁎C⁎-algebras and that the class of extremally rich C⁎C⁎-algebras with weak cancellation is closed under extensions. Moreover, we consider analogous properties which replace the group K1(A)K1(A) with the extremal K  -set Ke(A)Ke(A) as well as two versions of K0K0-surjectivity.