2019/11/18 by Rui Shi, Shi, Rui
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.1911.07696
openalex publication_date 2019/11/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In [10], Halmos proved an interesting result that the set of irreducible operators is dense in \mathcal B(\mathcal H) in the sense of Hilbert-Schmidt approximation. In a von Neumann algebra \mathcal M with separable predual, an operator a∈ \mathcal M is said to be irreducible in \mathcal M if W^*(a) is an irreducible subfactor of \mathcal M, i.e., W^*(a)'∩ \mathcal M=\mathbb C ⋅ I. In this paper, let Φ(⋅) be a \Vert⋅\Vert-dominating, unitarily invariant norm (see Definition 2.1), where by \Vert⋅\Vert we denote the operator norm. We prove that in every semifinite von Neumann factor \mathcal M with separable predual, if the norm Φ(⋅) satisfies a natural restriction introduced in (1.1), then irreducible operators are Φ(⋅)-norm dense in \mathcal M. In particular, the operator norm \Vert⋅\Vert and the max\\Vert⋅\Vert, \Vert⋅\Vertp\-norm (for each p>1) naturally satisfy the condition in (1.1), where τ is a faithful, normal, semifinite, tracial weight and \Vert x\Vertp=τ(|x|p)1/p for all x∈ \mathcal M ∩ Lp(\mathcal M,τ) (see [18, Preliminaries]). This can be viewed as a (stronger) analogue of a theorem of Halmos in [10], proved with different techniques developed in semifinite, properly infinite von Neumann factors. Meanwhile, for every \Vert⋅\Vert-dominating, unitarily invariant norm Φ(⋅), we develop another method to prove that each normal operator in \mathcal M is a sum of an irreducible operator in \mathcal M and an arbitrarily small Φ(⋅)-norm perturbation, where the Φ(⋅)-norm isn't restricted by (1.1). Particularly, the Φ(⋅)-norm can be the max\\Vert⋅\Vert, \Vert⋅\Vert1\-norm.