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

Abstract

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.