Sum of irreducible operators in von Neumann factors

Let M \mathcal {M} be a factor acting on a complex, separable Hilbert space H \mathcal {H} . An operator a ∈ M a\in \mathcal {M} is said to be irreducible in M \mathcal {M} if W ∗ ( a ) W^*(a) , the von Neumann subalgebra generated by a a in M \mathcal M , is an irreducible subfactor of M \mathcal {M} , i.e., W ∗ ( a ) ′ ∩ M = C I W^*(a)’\cap \mathcal {M}=\mathbb {C} I . In this note, we prove that each operator a ∈ M a\in \mathcal {M} is a sum of two irreducible operators in M \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.