Regular inclusions of simple unital $C^*$-algebras
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866918265767329792 |
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| author | Bakshi, Keshab Chandra Gupta, Ved Prakash |
| author_facet | Bakshi, Keshab Chandra Gupta, Ved Prakash |
| contents | We prove that an inclusion $\mathcal{B} \subset \mathcal{A}$ of simple unital $C^*$-algebras with a finite-index conditional expectation is regular if and only if there exists a finite group $G$ that admits a cocycle action $(α,σ)$ on the intermediate $C^*$-subalgebra $\mathcal{C}$ generated by $\mathcal{B}$ and its centralizer $\mathcal{C}_\mathcal{A}(\mathcal{B})$ such that $\mathcal{B}$ is outerly $α$-invariant and $(\mathcal{B} \subset \mathcal{A}) \cong ( \mathcal{B} \subset \mathcal{C}\rtimes^r_{α, σ} G)$. Prior to this characterization, we prove the existence of two-sided and unitary quasi-bases for the minimal conditional expectation of any such inclusion, and also show that such an inclusion has integer Watatani index and depth at most $2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_06959 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Regular inclusions of simple unital $C^*$-algebras Bakshi, Keshab Chandra Gupta, Ved Prakash Operator Algebras Functional Analysis We prove that an inclusion $\mathcal{B} \subset \mathcal{A}$ of simple unital $C^*$-algebras with a finite-index conditional expectation is regular if and only if there exists a finite group $G$ that admits a cocycle action $(α,σ)$ on the intermediate $C^*$-subalgebra $\mathcal{C}$ generated by $\mathcal{B}$ and its centralizer $\mathcal{C}_\mathcal{A}(\mathcal{B})$ such that $\mathcal{B}$ is outerly $α$-invariant and $(\mathcal{B} \subset \mathcal{A}) \cong ( \mathcal{B} \subset \mathcal{C}\rtimes^r_{α, σ} G)$. Prior to this characterization, we prove the existence of two-sided and unitary quasi-bases for the minimal conditional expectation of any such inclusion, and also show that such an inclusion has integer Watatani index and depth at most $2$. |
| title | Regular inclusions of simple unital $C^*$-algebras |
| topic | Operator Algebras Functional Analysis |
| url | https://arxiv.org/abs/2404.06959 |