Strongly self-absorbing 𝐶*-algebras

Source abstract

Say that a separable, unital C ∗ C^* -algebra D ≆ C \mathcal {D} \ncong \mathbb {C} is strongly self-absorbing if there exists an isomorphism φ : D → D ⊗ D \varphi : \mathcal {D} \to \mathcal {D} \otimes \mathcal {D} such that φ \varphi and i d D ⊗ 1 D \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 O 2 \mathcal {O}_2 , O ∞ \mathcal {O}_{\infty } , the UHF algebras of infinite type, the Jiang–Su algebra Z \mathcal {Z} and tensor products of O ∞ \mathcal {O}_{\infty } with UHF algebras of infinite type. Given a strongly self-absorbing C ∗ C^{*} -algebra D \mathcal {D} we characterise when a separable C ∗ C^* -algebra absorbs D \mathcal {D} tensorially (i.e., is D \mathcal {D} -stable), and prove closure properties for the class of separable D \mathcal {D} -stable C ∗ C^* -algebras. Finally, we compute the possible K K -groups and prove a number of classification results which suggest that the examples listed above are the only strongly self-absorbing C ∗ C^* -algebras.