Strongly self-absorbing C*-algebras

Abstract

Say that a separable, unital C*-algebra D is strongly self-absorbing if there exists an isomorphism $ϕ: D \to D \otimes D$ such that $ϕ$ and $id_D \otimes 1_D$ 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 Z and tensor products of $\Oh_{\infty}$ with UHF algebras of infinite type. Given a strongly self-absorbing C*-algebra D we characterise when a separable C*-algebra absorbs D tensorially (i.e., is D-stable), and prove closure properties for the class of separable 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.