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

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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 τ, a normal operator is an arbitrarily small ({·, ·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.