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

TL;DR

This paper extends the weak Dixmier property to type III von Neumann algebras with operator valued weights, leading to new structural insights and generalizations of Popa's intertwining technique without tracial assumptions.

Contribution

It introduces a relative weak Dixmier property for type III factors and applies it to generalize Popa's intertwining criterion and other structural theorems.

Findings

Proved the weak Dixmier property for positive elements with finite expectation.

Reformulated Popa's intertwining criterion without tracial assumptions.

Extended Ozawa's relative solidity theorem to type III factors.

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.