Irreducible operators in von Neumann algebras
Abstract
Let M \mathcal {M} be a separable von Neumann algebra with center Z ( M ) \mathcal {Z}(\mathcal {M}) . An operator T T in M \mathcal {M} is called irreducible if the von Neumann algebra W ∗ ( T ) W^*(T) generated by T T has trivial relative commutant, i.e., W ∗ ( T ) ′ ∩ M = Z ( M ) W^*(T)’\cap \mathcal {M}=\mathcal {Z}(\mathcal {M}) . In this paper, we show that irreducible operators in M \mathcal {M} form a norm-dense G δ G_\delta set, which is a generalization of Halmos’ theorem. Moreover, we prove that every operator in M \mathcal {M} is the sum of two irreducible operators, which is an analogue of Radjavi’s theorem.
Fields of Study & Subjects
Advanced Operator Algebra Research
Holomorphic and Operator Theory
Advanced Banach Space Theory
Publication details
- Journal / source
- Proceedings of the American Mathematical Society
- Volume
- 154
- Issue
- 11
- Pages
- 4721–4730
- Publication date
- 17 June 2026
- Language
- EN