Rokhlin actions and self-absorbing C*-algebras
Let A be a unital separable C∗-algebra, and D a K1-injective strongly selfabsorbing C∗-algebra. We show that if A is D-absorbing, then the crossed product of A by a compact second countable group or by Z or by R is D-absorbing as well, assuming the action satisfying a Rokhlin property. In the case of a compact Rokhlin action we prove a similar statement about approximate divisibility.
Strongly self-absorbing C*-algebras
Say that a separable, unital C*-algebra V ? C is strongly self absorbing if there exists an isomorphism V such that lx> ai>e approximately unitarily equivalent *-homomorphisms. We study this class of algebras, which includes the Cuntz algebras ?2, Ooo, the UHF algebras of infinite type, the Jiang-Su algebra Z and tensor products of ?00 with UHF algebras of infinite type. Given a strongly self-absorbing C*-algebra V we characterise when a separable C*-algebra absorbs V tensorially (i.e., is P-stable), and prove closure properties for the class of separable Testable C* algebras. Finally, we compute the possible If-groups and prove a number of classification results which suggest that the examples listed above are the only strongly self-absorbing C*-algebras.
Inclusions of simple $C^*$-algebras
We prove that if a conditional expectation from a simple $C^*$-algebra onto its $C^*$-subalgebra satisfies the Pimsner-Popa inequality, there exists a quasi-basis. As an application, we establish the Galois correspondence for outer actions of finite groups (and more generally finite dimensional Kac algebras) on simple $C^*$-algebras. We also introduce the notion of sectors for stable simple $C^*$-algebras and purely infinite simple $C^*$-algebras in the Cuntz standard form, and discuss several applications, including their relationship to $K$-theory.