Kadison's problem for type III subfactors and the bicentralizer conjecture
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866917831896989696 |
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| author | Marrakchi, Amine |
| author_facet | Marrakchi, Amine |
| contents | In 1967, Kadison asked "if $N$ is a subfactor of the factor $M$ for which $N' \cap M$ consists of scalars, will some maximal abelian *-subalgebra of $N$ be a maximal abelian subalgebra of $M$?". Generalizing a theorem of Popa in the type $\mathrm{II}$ case (1981), we solve Kadison's problem for all subfactors with expectation $N \subset M$ where $N$ is either a type $\mathrm{III}_λ$ factor with $0 \leq λ< 1$ or a type $\mathrm{III}_1$ factor that satisfies Connes's bicentralizer conjecture. Our solution is based on a new explicit formula for the bicentralizer algebras of arbitrary inclusions. This formula implies a type $\mathrm{III}$ analog of Popa's local quantization principle. We generalize Haaegrup's theorem from 1984 by connecting the relative bicentralizer conjecture to the Dixmier property. Finally, we prove this conjecture for a large class of inclusions and we prove an ergodicity theorem for the bicentralizer flow. We also give applications of our methods to $\mathrm{II}_1$ factors, including a new characterization of Ozawa's W*-Akemann-Ostrand property. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_15163 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Kadison's problem for type III subfactors and the bicentralizer conjecture Marrakchi, Amine Operator Algebras 46L10 (Primary) 46L36, 46L37, 46M07 (Secondary) In 1967, Kadison asked "if $N$ is a subfactor of the factor $M$ for which $N' \cap M$ consists of scalars, will some maximal abelian *-subalgebra of $N$ be a maximal abelian subalgebra of $M$?". Generalizing a theorem of Popa in the type $\mathrm{II}$ case (1981), we solve Kadison's problem for all subfactors with expectation $N \subset M$ where $N$ is either a type $\mathrm{III}_λ$ factor with $0 \leq λ< 1$ or a type $\mathrm{III}_1$ factor that satisfies Connes's bicentralizer conjecture. Our solution is based on a new explicit formula for the bicentralizer algebras of arbitrary inclusions. This formula implies a type $\mathrm{III}$ analog of Popa's local quantization principle. We generalize Haaegrup's theorem from 1984 by connecting the relative bicentralizer conjecture to the Dixmier property. Finally, we prove this conjecture for a large class of inclusions and we prove an ergodicity theorem for the bicentralizer flow. We also give applications of our methods to $\mathrm{II}_1$ factors, including a new characterization of Ozawa's W*-Akemann-Ostrand property. |
| title | Kadison's problem for type III subfactors and the bicentralizer conjecture |
| topic | Operator Algebras 46L10 (Primary) 46L36, 46L37, 46M07 (Secondary) |
| url | https://arxiv.org/abs/2308.15163 |