Sum of irreducible operators in von Neumann factors

Abstract:

Let $\mathcal {M}$ be a factor acting on a complex, separable Hilbert space $\mathcal {H}$. An operator $a\in \mathcal {M}$ is said to be irreducible in $\mathcal {M}$ if $W^*(a)$, the von Neumann subalgebra generated by $a$ in $\mathcal M$, is an irreducible subfactor of $\mathcal {M}$, i.e., $W^*(a)’\cap \mathcal {M}=\mathbb {C} I$. In this note, we prove that each operator $a\in \mathcal {M}$ is a sum of two irreducible operators in $\mathcal {M}$, which can be viewed as a natural generalization of a theorem in [Proc. Amer. Math. Soc. 21 (1969), pp. 251–252], with a completely different proof.