Irreducible operators in 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 $W^{\ast }(T)$ generated by $T$ has trivial relative commutant, i.e., $W^{\ast }(T{)}^{\prime }\cap \mathcal{M}=\mathcal{Z}(\mathcal{M})$. In this paper, we show that irreducible operators in $\mathcal{M}$ form a norm-dense $G_{\hat{I}}´$ 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.