Tusi Mathematical Research Group (TMRG)Advances in Operator Theory2538-225X3120180101Permanence of nuclear dimension for inclusions of unital $C^*$-algebras with the Rokhlin property1231364517710.22034/aot.1703-1145ENHiroyuki OsakaRitsumeikan UniversityTamotsu TeruyaJournal Article20170328Let $P subset A$ be an inclusion of unital $C^*$-algebras and $Ecolon A rightarrow P$ be a faithful conditional expectation of index finite type. Suppose that $E$ has the Rokhlin property. Then $dr(P) leq dr(A)$ and $dim_{nuc}(P) leq dim_{nuc}(A)$. This can be applied to Rokhlin actions of finite groups. We also show that under the same above assumption if $A$ is exact and pure, that is, the Cuntz semigroups $W(A)$ has strict comparison and is almost divisible, then $P$ and the basic contruction $C^*langle A, e_Prangle$ are also pure.http://www.aot-math.org/article_45177_609d8347a1a4c02639504efeafda0dce.pdf