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

In 1967, Kadison asked “if $N$ $N$ is a subfactor of the factor $M$ $M$ for which $N^{\prime }\cap M$ $N' \cap M$ consists of scalars, will some maximal abelian *-subalgebra of $N$ $N$ be a maximal abelian subalgebra of $M$ $M$ ?”. Generalizing a theorem of Popa in the type $\mathrm{II}$ $\mathrm {II}$ case (1981), we solve Kadison’s problem for all subfactors with expectation $N\subset M$ $N \subset M$ where $N$ $N$ is either a type ${\mathrm{III}}_{\lambda }$ $\mathrm {III}_{\lambda }$ factor with $0\le \lambda <1$ $0 \leq \lambda < 1$ or a type ${\mathrm{III}}_1$ $\mathrm {III}_{1}$ 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 $\mathrm{III}$ $\mathrm {III}$ 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.