The range of operators on von Neumann algebras

We prove that for every bounded linear operator T : X-->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 is an element of C the operator lambdaI-T fails to have dense range). In particular, and as an application, we obtain that such a space cannot support a topologically transitive operator.