The unital Ext-groups and classification of $C^{\ast }$-algebras

Abstract

The semigroups of unital extensions of separable $C^{\ast }$-algebras come in two flavours: a strong and a weak version. By the unital $\mathrm{Ext}$-groups, we mean the groups of invertible elements in these semigroups. We use the unital $\mathrm{Ext}$-groups to obtain $K$-theoretic classification of both unital and non-unital extensions of $C^{\ast }$-algebras, and in particular we obtain a complete $K$-theoretic classification of full extensions of UCT Kirchberg algebras by stable AF algebras.

31 pages. V2: Minor changes. To appear in Glasgow Math. J