Let be an inclusion of -unital -algebras with a finite index in the sense of Pimsner–Popa. Then we introduce the Rokhlin property for a conditional expectation from onto and show that if is simple and satisfies any of the property – listed in the below, and has the Rokhlin property, then so does :
-algebras that absorb a given strongly self-absorbing -algebra ;
-algebras of stable rank one;
-algebras of real rank zero;
-algebras of nuclear dimension at most , where ;
-algebras of decomposition rank at most , where ;
separable simple -algebras that are stably isomorphic to AF algebras;
separable simple -algebras that are stably isomorphic to AI algebras;
separable simple -algebras that are stably isomorphic to AT algebras;
separable simple -algebras that are stably isomorphic to sequential direct limits of one dimensional NCCW complexes;
separable -algebras with strict comparison of positive elements.
In particular, when is an action of a finite group on with the Rokhlin property in the sense of Nawata, the properties – are inherited to the fixed point algebra and the crossed product algebra from . In the case of a finite index inclusion of unital -algebras if the conditional expectation has the Rokhlin property in the sense of Izumi, the previous results except (11) are observed in previous works.
"The Rokhlin property for inclusions of $C^*$-algebras." Rocky Mountain J. Math. 50 (5) 1785 - 1792, October 2020. https://doi.org/10.1216/rmj.2020.50.1785