Products of irreducible operators in factors

Let $\mathcal{M}$ be a separable factor. An operator $T$ in $\mathcal{M}$ is said to be irreducible in $\mathcal{M}$ if the von Neumann algebra $W^{\ast }(T)$ generated by $T$ is an irreducible subfactor of $\mathcal{M}$, i.e., $W^{\ast }(T{)}^{\mathrm{\prime }}\cap \mathcal{M}=\mathbb{C}I$. In this paper, we show that every operator in a separable factor $\mathcal{M}$ is the product of two irreducible operators in $\mathcal{M}$, except the zero operator in factors of type ${\mathrm{I}}_{2n+1}$ for $n⩾1$. This may be viewed as a multiplicative analogue of Radjavi’s result which asserts that every operator on a separable Hilbert space is the sum of two irreducible operators.