Strongly self-absorbing C*-algebras

Abstract

Say that a separable, unital C*-algebra D is strongly self-absorbing if there exists an isomorphism $\varphi :D\to D\otimes D$ such that $\varphi$ and $i{d}_D\otimes 1_D$ are approximately unitarily equivalent $\ast$-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.

31 pages. Some minor errors corrected, table of reference updated. Exposition of Section 4 slightly improved. To appear in Trans. AMS

References in corpus (1)

• Rokhlin actions and self-absorbing C*-algebras

Cited by in corpus (1)

• On the classification of simple Z-stable C*-algebras with real rank zero and finite decomposition rank