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

Abstract

Let $P\subset A$ be an inclusion of $\sigma$-unital $C^{\ast }$-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 $\left(1\right)$–$\left(12\right)$ listed in the below, and $E$ has the Rokhlin property, then so does $P$:

1. simplicity;

2. nuclearity;

3. $C^{\ast }$-algebras that absorb a given strongly self-absorbing $C^{\ast }$-algebra $\mathcal{𝒟}$;

4. $C^{\ast }$-algebras of stable rank one;

5. $C^{\ast }$-algebras of real rank zero;

6. $C^{\ast }$-algebras of nuclear dimension at most $n$, where $n\in {\mathbb{Z}}^{+}$;

7. $C^{\ast }$-algebras of decomposition rank at most $n$, where $n\in {\mathbb{Z}}^{+}$;

8. separable simple $C^{\ast }$-algebras that are stably isomorphic to AF algebras;

9. separable simple $C^{\ast }$-algebras that are stably isomorphic to AI algebras;

10. separable simple $C^{\ast }$-algebras that are stably isomorphic to AT algebras;

11. separable simple $C^{\ast }$-algebras that are stably isomorphic to sequential direct limits of one dimensional NCCW complexes;

12. separable $C^{\ast }$-algebras with strict comparison of positive elements.

In particular, when $\alpha :G\to Aut\left(A\right)$ is an action of a finite group $G$ on $A$ with the Rokhlin property in the sense of Nawata, the properties $\left(1\right)$–$\left(12\right)$ are inherited to the fixed point algebra $A^{\alpha }$ and the crossed product algebra $A{⋊}_{\alpha }G$ from $A$. In the case of a finite index inclusion of unital $C^{\ast }$-algebras $P\subset A$ if the conditional expectation $E:A\to P$ has the Rokhlin property in the sense of Izumi, the previous results except (11) are observed in previous works.