Strongly self-absorbing $C^{*}$-algebras

Say that a separable, unital $C^*$-algebra $\mathcal{D} \ncong \mathbb{C}$ is strongly self-absorbing if there exists an isomorphism $\varphi: \mathcal{D} \to \mathcal{D} \otimes \mathcal{D}$ such that $\varphi$ and $\mathrm{id}_{\mathcal{D}} \otimes \mathbf{1}_{\mathcal{D}}$ are approximately unitarily equivalent $*$-homomorphisms. We study this class of algebras, which includes the Cuntz algebras $\mathcal{O}_2$, $\mathcal{O}_{\infty}$, the UHF algebras of infinite type, the Jiang-Su algebra $\mathcal{Z}$ and tensor products of $\mathcal{O}_{\infty}$ with UHF algebras of infinite type. Given a strongly self-absorbing $C^{*}$-algebra $\mathcal{D}$ we characterise when a separable $C^*$-algebra absorbs $\mathcal{D}$ tensorially (i.e., is $\mathcal{D}$-stable), and prove closure properties for the class of separable $\mathcal{D}$-stable $C^*$-algebras. Finally, we compute the possible $K$-groups and prove a number of classification results which suggest that the examples listed above are the only strongly self-absorbing $C^*$-algebras.