Irreducible operators in von Neumann algebras

2025/12/03 by Sukitha Adappa, Adappa, Sukitha, Minghui Ma +7
Mathematics · #FOS: Mathematics #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.2512.03448

Abstract

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

Related

• On irreducible operators in factor von Neumann algebras
• Products of irreducible operators in factors
• Normed ideal perturbation of irreducible operators in semifinite von Neumann factors
• Density of irreducible operators in the trace-class norm
• Divisible operators in von Neumann algebras