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

Abstract: 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 ( \sigma )-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.