Lattice of intermediate subalgebras
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2020
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| _version_ | 1866917175573348352 |
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| author | Bakshi, Keshab Chandra Gupta, Ved Prakash |
| author_facet | Bakshi, Keshab Chandra Gupta, Ved Prakash |
| contents | Analogous to subfactor theory, employing Watatani's notions of index and $C^*$-basic construction of certain inclusions of $C^*$-algebras, (a) we develop a Fourier theory (consisting of Fourier transforms, rotation maps and shift operators) on the relative commutants of any inclusion of simple unital $C^*$-algebras with finite Watatani index, and (b) we introduce the notions of interior and exterior angles between intermediate $C^*$-subalgebras of any inclusion of unital $C^*$-algebras admitting a finite index conditional expectation. Then, on the lines of [2], we apply these concepts to obtain a bound for the cardinality of the lattice of intermediate $C^*$-subalgebras of any irreducible inclusion as in (a), and improve Longo's bound for the cardinality of intermediate subfactors of an inclusion of type $III$ factors with finite index. Moreover, we also show that for a fairly large class of inclusions of finite von Neumann algebras, the lattice of intermediate von Neumann subalgebras is always finite. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2005_01049 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Lattice of intermediate subalgebras Bakshi, Keshab Chandra Gupta, Ved Prakash Operator Algebras Functional Analysis 47L40, 46L05, 46L37 Analogous to subfactor theory, employing Watatani's notions of index and $C^*$-basic construction of certain inclusions of $C^*$-algebras, (a) we develop a Fourier theory (consisting of Fourier transforms, rotation maps and shift operators) on the relative commutants of any inclusion of simple unital $C^*$-algebras with finite Watatani index, and (b) we introduce the notions of interior and exterior angles between intermediate $C^*$-subalgebras of any inclusion of unital $C^*$-algebras admitting a finite index conditional expectation. Then, on the lines of [2], we apply these concepts to obtain a bound for the cardinality of the lattice of intermediate $C^*$-subalgebras of any irreducible inclusion as in (a), and improve Longo's bound for the cardinality of intermediate subfactors of an inclusion of type $III$ factors with finite index. Moreover, we also show that for a fairly large class of inclusions of finite von Neumann algebras, the lattice of intermediate von Neumann subalgebras is always finite. |
| title | Lattice of intermediate subalgebras |
| topic | Operator Algebras Functional Analysis 47L40, 46L05, 46L37 |
| url | https://arxiv.org/abs/2005.01049 |