Strongly Self-absorbing C * -algebras

Say that a separable, unital C *-algebra D ≇ C is strongly self-absorbing if there exists an isomorphism ϕ : D → D ⊗ D such that ϕ and id D ⊗ 1 D are approximately unitarily equivalent *-homomorphisms. We study this class of algebras, which includes the Cuntz algebras O 2 , O∞, the UHF algebras of infinite type, the Jiang–Su algebra Z and tensor products of O∞ 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. Elliott's program to classify nuclear C*-algebras via K-theoretic invariants (see [7] for an introductory overview) has met with considerable success since his seminal classification of approximately finite-dimensional (AF) algebras via the scaled ordered K 0-group ([6]). An exhaustive list of the contributions to this pursuit would be prohibitively long, but salient works include A great variety of C *-algebras are studied by these authors, and, despite their apparent differences, all of them have been classified by K-theoretic invariants. Upon studying the literature related to Elliott's program, one finds that certain C *-algebras have been starting points for major stages of the classification program: UHF algebras in the stably finite case, and the Cuntz algebras in the purely infinite case. One can safely say that, among the Cuntz algebras, O 2 and O ∞ stand out; they are cornerstones of the Kirchberg–Phillips classification of simple purely infinite C *-algebras and of Kirchberg's classification of non-simple O 2-absorbing C *-algebras (in the case where said algebras satisfy the Universal Coefficients Theorem). There is evidence that the Jiang–Su algebra Z, which has recently come to the fore of the classification program, plays a role in the stably finite case similar to that of O ∞ in the purely infinite case (cf. [30], [36] and [42]). One might reasonably ask whether there is an abstract property which singles these algebras out from among their peers. UHF algebras (at least those of infinite type), O 2 , O ∞ and Z are all isomorphic to their tensor squares in a strong sense; for each algebra D from this list there exists …