A generalization of the Voiculescu theorem for normal operators in semifinite von Neumann algebras

In this paper, we provide a generalized version of the Voiculescu theorem for normal operators by showing that, in a von Neumann algebra with separable pre-dual and a faithful normal semifinite tracial weight $\tau$, a normal operator is an arbitrarily small $(\max\{\|\cdot\|, \Vert\cdot\Vert_{2}\})$-norm perturbation of a diagonal operator. Furthermore, in a countably decomposable, properly infinite von Neumann algebra with a faithful normal semifinite tracial weight, we prove that each self-adjoint operator can be diagonalized modulo norm ideals satisfying a natural condition.