On irreducible operators in factor von Neumann algebras

TL;DR

This paper proves that in a factor von Neumann algebra, the set of operators with irreducible generated subfactors is dense and topologically large, extending Halmos's classical result.

Contribution

It generalizes Halmos's theorem by showing the density and $G_\delta$-property of irreducible operators in factor von Neumann algebras.

Findings

Irreducible operators form a dense $G_\delta$ set in the algebra.

The result extends classical theorems to the setting of factor von Neumann algebras.

Provides a topological characterization of irreducible operators.

Abstract

Let $\mathcal M$ be a factor von Neumann algebra with separable predual and let $T\in \mathcal M$. We call $T$ an irreducible operator (relative to $\mathcal M$) if $W^*(T)$ is an irreducible subfactor of $\mathcal M$, i.e., $W^*(T)'\cap \mathcal M={\mathbb C} I$. In this note, we show that the set of irreducible operators in $\mathcal M$ is a dense $G_\delta$ subset of $\mathcal M$ in the operator norm. This is a natural generalization of a theorem of Halmos.