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Strongly self-absorbing -algebras

Authors:	Andrew S Toms Wilhelm Winter

Institution:	Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, Ontario, Canada M3J 1P3 ; Mathematisches Institut der Universität Münster, Einsteinstr. 62, D-48149 Münster, Germany

Abstract:	Say that a separable, unital -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 -algebra absorbs $\mathcal{D}$ tensorially (i.e., is $\mathcal{D}$ -stable), and prove closure properties for the class of separable $\mathcal{D}$ -stable -algebras. Finally, we compute the possible -groups and prove a number of classification results which suggest that the examples listed above are the only strongly self-absorbing -algebras.

Keywords:	Nuclear $C^*$-algebras K-theory classification

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