  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.
