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The semigroups of unital extensions of separable $C^\ast$-algebras come intwo 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 theunital $\mathrm{Ext}$-groups to obtain $K$-theoretic classification of bothunital and non-unital extensions of $C^\ast$-algebras, and in particular weobtain a complete $K$-theoretic classification of full extensions of UCTKirchberg algebras by stable AF algebras.

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