On irreducible operators in factor von Neumann algebras

Abstract

Let $\mathcal{M}$ be a factor von Neumann algebra with separable predual and let $T\in \mathcal{M}$. We call $T$ an irreducible operator (relative to $\mathcal{M}$) if $W^{\ast }(T)$ is an irreducible subfactor of $\mathcal{M}$, i.e., $W^{\ast }(T{)}^{\prime }\cap \mathcal{M}=\mathbb{C}I$. In this note, we show that the set of irreducible operators in $\mathcal{M}$ is a dense $G_{\hat{I}}´$ subset of $\mathcal{M}$ in the operator norm. This is a natural generalization of a theorem of Halmos.

4 pages