Kadison's problem for type III subfactors and the bicentralizer conjecture

• 成果类型: Article
• 署名作者: Marrakchi, Amine
• 署名单位: Ecole Normale Superieure de Lyon (ENS de LYON); Centre National de la Recherche Scientifique (CNRS)
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-024-01299-5
• 发表日期: 2025
• 页码: 79-163
• 关键词: OPERATOR-VALUED WEIGHTS; NEUMANN; EQUIVALENCE; FLOW; CLASSIFICATION; SUBALGEBRAS; ALGEBRAS; PRODUCTS; SPACE

摘要

In 1967, Kadison asked if N is a subfactor of the factor M for which N 'boolean AND Mconsistsof scalars, will some maximal abelian *-subalgebra of Nbe a maximal abelian sub-algebra of M?. Generalizing a theorem of Popa in the type II case (1981), we solve Kadison's problem for all subfactors with expectation N subset of M where Nis either atype III lambda factor with 0 <=lambda<1 or a type III 1 factor that satisfies Connes's bicentralizer conjecture. Our solution is based on a new explicit formula for the bicentralizeralgebras of arbitrary inclusions. This formula implies a type 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