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 \( E \colon M \to A \). We prove that every positive element \( x \in M \) with \( E(x) < \infty \) satisfies the weak Dixmier property relative to \( A \): the \( σ\)-weak closure of the convex hull of its unitary orbit under \( \mathcal{U}(A) \) intersects the relative commutant \( A' \cap M \). 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.