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

TL;DR

This paper solves Kadison's problem for certain type III subfactors by developing a new formula for bicentralizer algebras, extending Popa's theorem, and connecting the bicentralizer conjecture to the Dixmier property.

Contribution

The authors provide a new explicit formula for bicentralizer algebras, generalize Popa's local quantization principle, and prove the bicentralizer conjecture for a broad class of inclusions.

Findings

Solved Kadison's problem for type III subfactors with expectation.

Established a type III analog of Popa's local quantization principle.

Proved the relative bicentralizer conjecture for a large class of inclusions.

Abstract

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 $\mathrm{III}_\lambda$ factor with $0 \leq \lambda < 1$ 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.