Journal of Operator Theory

Volume 93, Issue 1, Winter 2025 pp. 291-312.

Approximate representability of finite abelian group actions on the Razak-Jacelon algebra

Authors: Norio Nawata
Author institution: Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Yamadaoka 1-5, Suita, Osaka 565-0871, Japan

Summary: Let $A$ be a simple separable nuclear monotracial $C^*$-algebra, and let $\alpha$ be an outer action of a finite abelian group $\Gamma$ on $A$. In this paper, we show that $\alpha\otimes \mathrm{id}_{\mathcal{W}}$ on $A\otimes\mathcal{W}$ is approximately representable if and only if the characteristic invariant of $\widetilde{\alpha}$ is trivial, where $\mathcal{W}$ is the Razak--Jacelon algebra and $\widetilde{\alpha}$ is the induced action on the injective II$_1$ factor $\pi_{\tau_{A}}(A)^{''}$. As an application of this result, we classify such actions up to conjugacy and cocycle conjugacy. We also construct the model actions.

DOI: http://dx.doi.org/10.7900/jot.2023may05.2443
Keywords: Razak-Jacelon algebra, approximate representability, Rohlin property, Kirchberg's central sequence $C^*$-algebra

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