Irreducible operators in von Neumann algebras

TL;DR

The paper extends Halmos' irreducibility results to separable von Neumann algebras with nontrivial center by proving that irreducible operators form a norm-dense $G_\delta$ set and that every operator is a sum of two irreducible operators. It introduces a perturbation framework: small, ideal-perturbations can yield irreducible operators, and this underpins the density result. The authors develop central-support techniques and block-decomposition methods to construct irreducible elements and to realize operator sums, including matrix-augmented and direct-sum contexts. Together, these results provide a robust generalization of classical irreducibility phenomena to the setting of von Neumann algebras and raise questions about strongly irreducible operators and related decompositions.

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 $G_δ$ 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.

Theorems & Definitions (26)

• Definition 1.1
• Theorem 1.2
• Theorem 1.4
• Definition 1.5
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof