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Normed ideal perturbation of irreducible operators in semifinite von Neumann factors

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 $Φ(\cdot)$ 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 $Φ(\cdot)$ satisfies a natural restriction introduced in (1.1), then irreducible operators are $Φ(\cdot)$-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 $τ$ is a faithful, normal, semifinite, tracial weight and $\Vert x\Vert_p=τ(|x|^p)^{1/p}$ for all $x\in \mathcal M \cap L^{p}(\mathcal M,τ)$ (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 $\Vert\cdot\Vert$-dominating, unitarily invariant norm $Φ(\cdot)$, 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 $Φ(\cdot)$-norm perturbation, where the $Φ(\cdot)$-norm isn't restricted by (1.1). Particularly, the $Φ(\cdot)$-norm can be the $\max\{\Vert\cdot\Vert, \Vert\cdot\Vert_1\}$-norm.