On \(\mathfrak{L}_{\infty}\)-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 \(\mathscr{B}(H)\) factors through an \(\mathfrak{L}_{\infty}\) -space only if \(M\) is a finite type \(\mathrm{I}\) algebra