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

2017/06/29 by Qihui Li, Junhao Shen, Li, Qihui +3 · 3 citations
Mathematics · #47C15 #Advanced Operator Algebra Research #FOS: Mathematics #Holomorphic and Operator Theory #Operator Algebras (math.OA) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1706.09522

openalex publication_date 2017/06/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 (max\‖⋅‖, \Vert⋅\Vert₂\)-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.

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