The Rohlin property for inclusions of $C^{\ast }$-algebras with a finite Watatani index

We introduce notions of the Rohlin property and the approximate representability for inclusions of unital $C^{\ast }$-algebras. We investigate a dual relation between the Rohlin property and the approximate representability. We prove that a number of classes of unital $C^{\ast }$-algebras are closed under inclusions with the Rohlin property, including: AF algebras, AI algebras, AT algebras, and related classes characterized by direct limit decomposition using semiprojective building blocks. $C^{\ast }$-algebras with stable rank one. $C^{\ast }$-algebras with real rank zero.