Approximate representability of finite abelian group actions on the Razak-Jacelon algebra
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866910639349301248 |
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| author | Nawata, Norio |
| author_facet | Nawata, Norio |
| contents | Let $A$ be a simple separable nuclear monotracial C$^*$-algebra, and let $α$ be an outer action of a finite abelian group $Γ$ on $A$. In this paper, we show that $α\otimes \mathrm{id}_{\mathcal{W}}$ on $A\otimes\mathcal{W}$ is approximately representable if and only if the characteristic invariant of $\tildeα$ is trivial, where $\mathcal{W}$ is the Razak-Jacelon algebra and $\tildeα$ is the induced action on the injective II$_1$ factor $π_{τ_{A}}(A)^{''}$. As an application of this result, we classify such actions up to conjugacy and cocycle conjugacy. In particular, we show the following: Let $A$ and $B$ be simple separable nuclear monotracial C$^*$-algebras, and let $α$ and $β$ be outer actions of a finite abelian group $Γ$ on $A$ and $B$, respectively. Assume that the characteristic invariants of $\tildeα$ and $\tildeβ$ are trivial. Then $α\otimes \mathrm{id}_{\mathcal{W}}$ and $β\otimes \mathrm{id}_{\mathcal{W}}$ are conjugate (resp. cocycle conjugate) if and only if $\tildeα$ on $π_{τ_{A}}(A)^{''}$ and $\tildeβ$ on $π_{τ_{B}}(B)^{''}$ are conjugate (resp. cocycle conjugate). We also construct the model actions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_10550 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Approximate representability of finite abelian group actions on the Razak-Jacelon algebra Nawata, Norio Operator Algebras Primary 46L55, Secondary 46L35, 46L40 Let $A$ be a simple separable nuclear monotracial C$^*$-algebra, and let $α$ be an outer action of a finite abelian group $Γ$ on $A$. In this paper, we show that $α\otimes \mathrm{id}_{\mathcal{W}}$ on $A\otimes\mathcal{W}$ is approximately representable if and only if the characteristic invariant of $\tildeα$ is trivial, where $\mathcal{W}$ is the Razak-Jacelon algebra and $\tildeα$ is the induced action on the injective II$_1$ factor $π_{τ_{A}}(A)^{''}$. As an application of this result, we classify such actions up to conjugacy and cocycle conjugacy. In particular, we show the following: Let $A$ and $B$ be simple separable nuclear monotracial C$^*$-algebras, and let $α$ and $β$ be outer actions of a finite abelian group $Γ$ on $A$ and $B$, respectively. Assume that the characteristic invariants of $\tildeα$ and $\tildeβ$ are trivial. Then $α\otimes \mathrm{id}_{\mathcal{W}}$ and $β\otimes \mathrm{id}_{\mathcal{W}}$ are conjugate (resp. cocycle conjugate) if and only if $\tildeα$ on $π_{τ_{A}}(A)^{''}$ and $\tildeβ$ on $π_{τ_{B}}(B)^{''}$ are conjugate (resp. cocycle conjugate). We also construct the model actions. |
| title | Approximate representability of finite abelian group actions on the Razak-Jacelon algebra |
| topic | Operator Algebras Primary 46L55, Secondary 46L35, 46L40 |
| url | https://arxiv.org/abs/2302.10550 |