Abstract
Let $P \subset A$ be an inclusion of unital $C^*$-algebras and $E: A \rightarrow P$ be a faithful conditional expectation of index finite type. Suppose that $E$ has the Rokhlin property. Then $\mathrm{dr}(P) \leq \mathrm{dr}(A)$ 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.
Citation
Hiroyuki Osaka. Tamotsu Teruya. "Permanence of nuclear dimension for inclusions of unital $C^*$-algebras with the Rokhlin property." Adv. Oper. Theory 3 (1) 123 - 136, Winter 2018. https://doi.org/10.22034/aot.1703-1145
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