# On the weak relative Dixmier property

## TL;DR

This paper proves that all inclusions of von Neumann algebras with a faithful normal conditional expectation possess the weak relative Dixmier property, resolving a question posed by Popa.

## Contribution

It establishes the weak relative Dixmier property for all such inclusions, using an improved version of Ellis' lemma for compact convex semigroups.

## Key findings

- All inclusions of von Neumann algebras with a faithful normal conditional expectation have the weak relative Dixmier property.
- Provides an affirmative answer to Popa's question from 1999.
- Introduces an improved version of Ellis' lemma for compact convex semigroups.

## Abstract

We show that every inclusion of von Neumann algebras with a faithful normal conditional expectation has the weak relative Dixmier property. This answers a question of Popa \cite{Po99}. The proof uses an improvement of Ellis' lemma for compact convex semigroup.