Crossed product splitting of intermediate operator algebras via 2-cocycles
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866912903748124672 |
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| author | Suzuki, Yuhei |
| author_facet | Suzuki, Yuhei |
| contents | We investigate the C*-algebra inclusions $B \subset A \rtimes_{\rm r} Γ$ arising from inclusions $B \subset A$ of $Γ$-C*-algebras. The main result shows that, when $B \subset A$ is C*-irreducible in the sense of Rørdam, and is centrally $Γ$-free in the sense of the author, then after tensoring with the Cuntz algebra $\mathcal{O}_2$, all intermediate C*-algebras $B \subset C\subset A \rtimes_{\rm r} Γ$ enjoy a natural crossed product splitting \[\mathcal{O}_2\otimes C=(\mathcal{O}_2 \otimes D) \rtimes_{{\rm r}, γ, \mathfrak{w}} Λ\] for $D:= C \cap A$, some $Λ<Γ$, and a subsystem $(γ, \mathfrak{w})$ of a unitary perturbed cocycle action $Λ\curvearrowright \mathcal{O}_2\otimes A$. As an application, we give a new Galois's type theorem for the Bisch--Haagerup type inclusions \[A^K \subset A\rtimes_{\rm r} Γ\] for actions of compact-by-discrete groups $K \rtimes Γ$ on simple C*-algebras.
Due to a K-theoretical obstruction, the operation $\mathcal{O}_2\otimes -$ is necessary to obtain the clean splitting. Also, in general 2-cocycles $\mathfrak{w}$ appearing in the splitting cannot be removed even further tensoring with any unital (cocycle) action. We show them by examples, which further show that $\mathcal{O}_2$ is a minimal possible choice.
We also establish a von Neumann algebra analogue, where $\mathcal{O}_2$ is replaced by the type I factor $\mathbb{B}(\ell^2(\mathbb{N}))$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_00304 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Crossed product splitting of intermediate operator algebras via 2-cocycles Suzuki, Yuhei Operator Algebras Dynamical Systems Group Theory Primary 46L55, Secondary 46L35 We investigate the C*-algebra inclusions $B \subset A \rtimes_{\rm r} Γ$ arising from inclusions $B \subset A$ of $Γ$-C*-algebras. The main result shows that, when $B \subset A$ is C*-irreducible in the sense of Rørdam, and is centrally $Γ$-free in the sense of the author, then after tensoring with the Cuntz algebra $\mathcal{O}_2$, all intermediate C*-algebras $B \subset C\subset A \rtimes_{\rm r} Γ$ enjoy a natural crossed product splitting \[\mathcal{O}_2\otimes C=(\mathcal{O}_2 \otimes D) \rtimes_{{\rm r}, γ, \mathfrak{w}} Λ\] for $D:= C \cap A$, some $Λ<Γ$, and a subsystem $(γ, \mathfrak{w})$ of a unitary perturbed cocycle action $Λ\curvearrowright \mathcal{O}_2\otimes A$. As an application, we give a new Galois's type theorem for the Bisch--Haagerup type inclusions \[A^K \subset A\rtimes_{\rm r} Γ\] for actions of compact-by-discrete groups $K \rtimes Γ$ on simple C*-algebras. Due to a K-theoretical obstruction, the operation $\mathcal{O}_2\otimes -$ is necessary to obtain the clean splitting. Also, in general 2-cocycles $\mathfrak{w}$ appearing in the splitting cannot be removed even further tensoring with any unital (cocycle) action. We show them by examples, which further show that $\mathcal{O}_2$ is a minimal possible choice. We also establish a von Neumann algebra analogue, where $\mathcal{O}_2$ is replaced by the type I factor $\mathbb{B}(\ell^2(\mathbb{N}))$. |
| title | Crossed product splitting of intermediate operator algebras via 2-cocycles |
| topic | Operator Algebras Dynamical Systems Group Theory Primary 46L55, Secondary 46L35 |
| url | https://arxiv.org/abs/2406.00304 |