Normed ideal perturbation of irreducible operators in semifinite von Neumann factors

DOI10.1007/S00020-021-02654-4arXiv1911.07696MaRDI QIDQ6329404

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Publication date: 18 November 2019

Abstract: In [10], Halmos proved an interesting result that the set of irreducible operators is dense in $mathcalB(mathcalH)$ in the sense of Hilbert-Schmidt approximation. In a von Neumann algebra $mathcalM$ with separable predual, an operator $ainmathcalM$ is said to be {irreducible in} $mathcalM$ if $W^{\ast }(a)$ is an irreducible subfactor of $mathcalM$, i.e., $W^{\ast }(a{)}^{\prime }capmathcalM=mathbbCcdotI$. In this paper, let $Phi(cdot)$ be a $VertcdotVert$-dominating, unitarily invariant norm (see Definition 2.1), where by $VertcdotVert$ we denote the operator norm. We prove that in every semifinite von Neumann factor $mathcalM$ with separable predual, if the norm $Phi(cdot)$ satisfies a natural restriction introduced in (1.1), then irreducible operators are $Phi(cdot)$-norm dense in $mathcalM$. In particular, the operator norm $VertcdotVert$ and the $maxVertcdotVert,VertcdotVer{t}_p$-norm (for each $p>1$) naturally satisfy the condition in (1.1), where $au$ is a faithful, normal, semifinite, tracial weight and $VertxVer{t}_p=au(|x{|}^p{)}^{1/p}$ for all $xinmathcalMcap{L}^p(mathcalM,au)$ (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 $VertcdotVert$-dominating, unitarily invariant norm $Phi(cdot)$, we develop another method to prove that each normal operator in $mathcalM$ is a sum of an irreducible operator in $mathcalM$ and an arbitrarily small $Phi(cdot)$-norm perturbation, where the $Phi(cdot)$-norm isn't restricted by (1.1). Particularly, the $Phi(cdot)$-norm can be the $maxVertcdotVert,VertcdotVer{t}_1$-norm.

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