Normed ideal perturbation of irreducible operators in semifinite von Neumann factors

TL;DR

This paper proves that in semifinite von Neumann factors, irreducible operators are dense with respect to certain norms, extending Halmos's classical result to a broader operator algebra setting.

Contribution

It establishes the density of irreducible operators in semifinite von Neumann factors under various unitarily invariant norms, generalizing prior results and introducing new approximation techniques.

Findings

Irreducible operators are dense in the operator norm and related norms in semifinite von Neumann factors.

Normal operators can be approximated by irreducible operators plus small perturbations in these norms.

The results extend Halmos's theorem to the setting of semifinite von Neumann algebras with new methods.

Abstract

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 $a\in \mathcal M$ is said to be {irreducible in} $\mathcal M$ if $W^*(a)$ is an irreducible subfactor of $\mathcal M$, i.e., $W^*(a)'\cap \mathcal M={\mathbb C} \cdot I$. In this paper, let $\Phi(\cdot)$ be a $\Vert\cdot\Vert$-dominating, unitarily invariant norm (see Definition 2.1), where by $\Vert\cdot\Vert$ we denote the operator norm. We prove that in every semifinite von Neumann factor $\mathcal M$ 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 $\mathcal M$. In particular, the operator norm $\Vert\cdot\Vert$ and the $\max\{\Vert\cdot\Vert, \Vert\cdot\Vert_p\}$-norm (for each $p>1$) naturally satisfy the condition in (1.1), where $\tau$ is a faithful, normal, semifinite, tracial weight and $\Vert x\Vert_p=\tau(|x|^p)^{1/p}$ for all $x\in \mathcal M \cap L^{p}(\mathcal M,\tau)$ (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 $\Vert\cdot\Vert$-dominating, unitarily invariant norm $\Phi(\cdot)$, 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 $\Phi(\cdot)$-norm perturbation, where the $\Phi(\cdot)$-norm isn't restricted by (1.1). Particularly, the $\Phi(\cdot)$-norm can be the $\max\{\Vert\cdot\Vert, \Vert\cdot\Vert_1\}$-norm.