Weak relative Dixmier property and Popa's intertwining technique for type III subfactors

Let $A\subset M$ be an inclusion of von Neumann algebras equipped with a faithful normal semifinite operator valued weight $E\text{\hspace{0.17em}⁣}:M\to A$. We prove that every positive element $x\in M$ with $E(x)<\mathrm{\infty }$ satisfies the weak Dixmier property relative to $A$: the $\sigma$-weak closure of the convex hull of its unitary orbit under $\mathcal{U}(A)$ intersects the relative commutant $A^{\mathrm{\prime }}\cap M$. This extends Marrakchi’s result for the case of conditional expectations. We apply this result to obtain new structural theorems for type III factors, including a reformulation of Popa’s intertwining criterion without tracial assumptions, an extension of Ozawa’s relative solidity theorem to the type III setting, and a Galois-type correspondence for crossed products by totally disconnected groups. The last result resolves a question posed by Boutonnet and Brothier regarding the structure of intermediate subfactors.