Topologies on central extensions of von Neumann algebras

Given a von Neumann algebra M, we consider the central extension E(M) of M. We introduce the topology t _c(M) on E(M) generated by a center-valued norm and prove that it coincides with the topology of local convergence in measure on E(M) if and only if M does not have direct summands of type II. We also show that t _c(M) restricted to the set E(M) _h of self-adjoint elements of E(M) coincides with the order topology on E(M) _h if and only if M is a σ-finite type I _fin von Neumann algebra.