Irreducible operators in von Neumann algebras

TL;DR

This paper studies irreducible operators within separable von Neumann algebras, proving their density and that any operator can be expressed as a sum of two irreducibles, extending classical theorems.

Contribution

It generalizes Halmos' theorem by showing irreducible operators are dense and constructs a decomposition of any operator into two irreducibles.

Findings

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

Every operator can be written as a sum of two irreducible operators.

Extension of classical theorems to the setting of von Neumann algebras.

Abstract

Let $\mathcal{M}$ be a separable von Neumann algebra with center $\mathcal{Z}(\mathcal{M})$. An operator $T$ in $\mathcal{M}$ is called irreducible if the von Neumann algebra $W^*(T)$ generated by $T$ has trivial relative commutant, i.e., $W^*(T)'\cap\mathcal{M}=\mathcal{Z}(\mathcal{M})$. In this paper, we show that irreducible operators in $\mathcal{M}$ form a norm-dense $G_\delta$ set, which is a generalization of Halmos' theorem. Moreover, we prove that every operator in $\mathcal{M}$ is the sum of two irreducible operators, which is an analogue of Radjavi's theorem.