  In the study on the diagonality of an $n$-tuple $\alpha=(\alpha(j))_{j=1}^n$
of commuting self-adjoint operators modulo a given $n$-tuple
$\Phi=(\mathcal{J}_1,\ldots,\mathcal{J}_n)$ of normed ideals in $B(H)$,
Voiculescu introduced the notion of quasicentral modulus $k_{\Phi}(\alpha)$ and
proved that $\alpha$ is diagonal modulo $(\mathcal{J}_1,\ldots,\mathcal{J}_n)$
if and only if $k_{\Phi}(\alpha)=0.$ We prove that the same assertion holds
true when $B(H)$ is replaced with a $\sigma$-finite semifinite von Neumann
algebra $\mathcal{M}$, and $\mathcal{J}_1,\ldots,\mathcal{J}_n$ are replaced
with symmetric spaces $E_1(\mathcal{M}),\ldots,E_n(\mathcal{M})$ associated
with $\mathcal{M}.$
