Classification of ${\mathcal{O}}_{\mathrm{\infty }}$-stable $C^{\ast }$-algebras

I present a proof of Kirchberg’s classification theorem: two separable, nuclear, ${\mathcal{O}}_{\mathrm{\infty }}$-stable $C^{\ast }$-algebras are stably isomorphic if and only if they are ideal-related $KK$-equivalent. In particular, this provides a more elementary proof of the Kirchberg–Phillips theorem which is isolated in the paper to increase readability of this important special case.