Advances in Operator Theory

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

Hiroyuki Osaka and Tamotsu Teruya

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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.

Article information

Adv. Oper. Theory, Volume 3, Number 1 (2018), 123-136.

Received: 28 March 2017
Accepted: 27 April 2017
First available in Project Euclid: 5 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46L55: Noncommutative dynamical systems [See also 28Dxx, 37Kxx, 37Lxx, 54H20]
Secondary: 46L35: Classifications of $C^*$-algebras

Rokhlin property $C^*$-index nuclear dimension


Osaka, Hiroyuki; Teruya, Tamotsu. Permanence of nuclear dimension for inclusions of unital $C^*$-algebras with the Rokhlin property. Adv. Oper. Theory 3 (2018), no. 1, 123--136. doi:10.22034/aot.1703-1145.

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