The Rokhlin property for inclusions of $C^*$-algebras

Abstract

Let P⊂A be an inclusion of σ-unital C∗-algebras with a finite index in the sense of Pimsner–Popa. Then we introduce the Rokhlin property for a conditional expectation E from A onto P and show that if A is simple and satisfies any of the property (1)–(12) listed in the below, and E has the Rokhlin property, then so does P: simplicity; nuclearity; C∗-algebras that absorb a given strongly self-absorbing C∗-algebra 𝒟; C∗-algebras of stable rank one; C∗-algebras of real rank zero; C∗-algebras of nuclear dimension at most n, where n∈ℤ+; C∗-algebras of decomposition rank at most n, where n∈ℤ+; separable simple C∗-algebras that are stably isomorphic to AF algebras; separable simple C∗-algebras that are stably isomorphic to AI algebras; separable simple C∗-algebras that are stably isomorphic to AT algebras; separable simple C∗-algebras that are stably isomorphic to sequential direct limits of one dimensional NCCW complexes; separable C∗-algebras with strict comparison of positive elements. In particular, when α:G→ Aut(A) is an action of a finite group G on A with the Rokhlin property in the sense of Nawata, the properties (1)–(12) are inherited to the fixed point algebra Aα and the crossed product algebra A⋊αG from A. In the case of a finite index inclusion of unital C∗-algebras P⊂A if the conditional expectation E:A→P has the Rokhlin property in the sense of Izumi, the previous results except (11) are observed in previous works.