The range of operators on von Neumann algebras

We prove that for every bounded linear operator $T:X\to X$, where $X$ is a non-reflexive quotient of a von Neumann algebra, the point spectrum of $T^*$ is non-empty (i.e. for some $\lambda\in\mathbb C$ the operator $\lambda I-T$ fails to have dense range.) In particular, and as an application, we obtain that such a space cannot support a topologically transitive operator.