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

2023/08/29 by Marrakchi, Amine · 2 citations
#46L10 (Primary) 46L36 #46L37 #46M07 (Secondary) #FOS: Mathematics #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.2308.15163

Abstract

In 1967, Kadison asked "if N is a subfactor of the factor M for which N' ∩ 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 II case (1981), we solve Kadison's problem for all subfactors with expectation N ⊂ M where N is either a type III_λ factor with 0 ≤ λ< 1 or a type III₁ 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 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 II₁ factors, including a new characterization of Ozawa's W*-Akemann-Ostrand property.

Cited by

• Cocycle perturbations and ergodicity for actions on type III factors
• Compressible subalgebras in II₁ factors

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