Depth 2 inclusions of simple $C^*$-algebras and their weak $C^*$-Hopf algebra symmetries
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866917265751932928 |
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| author | Pal, Biplab |
| author_facet | Pal, Biplab |
| contents | Let $B \subset A$ be a depth $2$ inclusion of simple unital $C^*$-algebras with a conditional expectation of index-finite type. We show that the second relative commutant $B' \cap A_1$ carries a canonical structure of a weak $C^*$-Hopf algebra. Furthermore, we construct an action of this weak $C^*$-Hopf algebra on $A$ for which $B$ is precisely the fixed-point subalgebra, and we prove that the first basic construction $A_1$ is isomorphic to the crossed product $A \rtimes (B' \cap A_1)$. This provides a $C^*$-algebraic counterpart of the duality between depth $2$ subfactors and weak Hopf algebra symmetry, extending the Ocneanu-Nikshych-Vainerman theory beyond the $II_1$ factor setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_19954 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Depth 2 inclusions of simple $C^*$-algebras and their weak $C^*$-Hopf algebra symmetries Pal, Biplab Operator Algebras 46L37, 47L40, 46L05, 46L35, 46L67 Let $B \subset A$ be a depth $2$ inclusion of simple unital $C^*$-algebras with a conditional expectation of index-finite type. We show that the second relative commutant $B' \cap A_1$ carries a canonical structure of a weak $C^*$-Hopf algebra. Furthermore, we construct an action of this weak $C^*$-Hopf algebra on $A$ for which $B$ is precisely the fixed-point subalgebra, and we prove that the first basic construction $A_1$ is isomorphic to the crossed product $A \rtimes (B' \cap A_1)$. This provides a $C^*$-algebraic counterpart of the duality between depth $2$ subfactors and weak Hopf algebra symmetry, extending the Ocneanu-Nikshych-Vainerman theory beyond the $II_1$ factor setting. |
| title | Depth 2 inclusions of simple $C^*$-algebras and their weak $C^*$-Hopf algebra symmetries |
| topic | Operator Algebras 46L37, 47L40, 46L05, 46L35, 46L67 |
| url | https://arxiv.org/abs/2511.19954 |