Classification of amenable group actions on C*-algebras and its applications

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Let P < A be an inclusion of unital C*--algebras with a finite Watatani Index. We show that if an inclusion P < A has the Rokhlin property in the sense of Kodaka, Osaka, Teruya, then the nuclear dimension of A is greater than or equal to that of P. This is a partial answer to a question by Winter-Zacharias. Moreover, under an inclusion P < A has the tracial Rokhlin property in the sense of Osaka and Teruya, if A is the Jiang-Su stable, then so is P.