Splitting of Tensor Products and Intermediate Factor Theorem: Continuous Version
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912434414944256 |
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| author | Amrutam, Tattwamasi Jiang, Yongle |
| author_facet | Amrutam, Tattwamasi Jiang, Yongle |
| contents | Let $G$ be a discrete group. Given unital $G$-$C^*$-algebras $\mathcal{A}$ and $\mathcal{B}$, we give an abstract condition under which every $G$-subalgebra $\mathcal{C}$ of the form $\mathcal{A}\subset \mathcal{C}\subset \mathcal{A}\otimes_{\text{min}}\mathcal{B}$ is a tensor product. This generalizes the well-known splitting results in the context of $C^*$-algebras by Zacharias and Zsido. As an application, we prove a topological version of the Intermediate Factor theorem. When a product group $G=Γ_1\timesΓ_2$ acts (by a product action) on the product of corresponding $Γ_i$-boundaries $\partialΓ_i$, using the abstract condition, we show that every intermediate subalgebra $C(X)\subset\mathcal{C}\subset C(X)\otimes_{\text{min}}C(\partialΓ_1\times \partialΓ_2)$ is a tensor product (under some additional assumptions on $X$). This can be considered as a topological version of the Intermediate Factor theorem. We prove that our assumptions are necessary and cannot generally be relaxed. We also introduce the notion of a uniformly rigid action for $C^*$-algebras and use it to give various classes of inclusions $\mathcal{A}\subset \mathcal{A}\otimes_{\text{min}}\mathcal{B}$ for which every invariant intermediate algebra is a tensor product. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_08635 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Splitting of Tensor Products and Intermediate Factor Theorem: Continuous Version Amrutam, Tattwamasi Jiang, Yongle Operator Algebras Dynamical Systems Functional Analysis Primary 46M05, 46L06, Secondary 37B05, 46L45 Let $G$ be a discrete group. Given unital $G$-$C^*$-algebras $\mathcal{A}$ and $\mathcal{B}$, we give an abstract condition under which every $G$-subalgebra $\mathcal{C}$ of the form $\mathcal{A}\subset \mathcal{C}\subset \mathcal{A}\otimes_{\text{min}}\mathcal{B}$ is a tensor product. This generalizes the well-known splitting results in the context of $C^*$-algebras by Zacharias and Zsido. As an application, we prove a topological version of the Intermediate Factor theorem. When a product group $G=Γ_1\timesΓ_2$ acts (by a product action) on the product of corresponding $Γ_i$-boundaries $\partialΓ_i$, using the abstract condition, we show that every intermediate subalgebra $C(X)\subset\mathcal{C}\subset C(X)\otimes_{\text{min}}C(\partialΓ_1\times \partialΓ_2)$ is a tensor product (under some additional assumptions on $X$). This can be considered as a topological version of the Intermediate Factor theorem. We prove that our assumptions are necessary and cannot generally be relaxed. We also introduce the notion of a uniformly rigid action for $C^*$-algebras and use it to give various classes of inclusions $\mathcal{A}\subset \mathcal{A}\otimes_{\text{min}}\mathcal{B}$ for which every invariant intermediate algebra is a tensor product. |
| title | Splitting of Tensor Products and Intermediate Factor Theorem: Continuous Version |
| topic | Operator Algebras Dynamical Systems Functional Analysis Primary 46M05, 46L06, Secondary 37B05, 46L45 |
| url | https://arxiv.org/abs/2408.08635 |