Diagonality modulo symmetric spaces in semifinite von Neumann algebras

In the study on the diagonality of an n-tuple α=(α(j))ⱼ₌₁ⁿ of commuting self-adjoint operators modulo a given n-tuple Φ=(J₁,…,Jₙ) of normed ideals in $B(H)$, Voiculescu introduced the notion of quasicentral modulus kΦ(α) and proved that α is diagonal modulo (J₁,…,Jₙ) if and only if kΦ(α)=0. We prove that the same assertion holds true when $B(H)$ is replaced with a σ-finite semifinite von Neumann algebra M, and J₁,…,Jₙ are replaced with symmetric spaces E₁(M),…,Eₙ(M) associated with M.