On ${\mathfrak{L}}_{\mathrm{\infty }}$-liftings of the Gelfand-Naimark morphism

Let $M$ be a $W^{\ast }$-algebra acting on a separable complex Hilbert space $H$. We show that the inclusion of $M$ into $\mathcal{B}(H)$ factors through an ${\mathfrak{L}}_{\mathrm{\infty }}$-space only if $M$ is a finite type $\mathrm{I}$ algebra.