Permanence of nuclear dimension for inclusions of unital $C^*$-algebras with the Rokhlin property

Document Type: Article dedicated to Uffe Haagerup


Ritsumeikan University


Let $P \subset A$ be an inclusion of unital $C^*$-algebras and $E\colon A \rightarrow P$ 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 $C^*\langle A, e_P\rangle$ are also pure.