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 $W^{\ast }(a)$ is an irreducible subfactor of $\mathcal{M}$, i.e., $W^*(a)'\cap \mathcal M={\mathbb C} \cdot I$\mathrm{.}Inthispaper,let$\Phi(\cdot)$bea$\Vert\cdot\Vert$-dominating, unitarily invariant norm (see Definition 2.1), where by $\mathrm{\parallel }\cdot \mathrm{\parallel }$ we denote the operator norm. We prove that in every semifinite von Neumann factor $\mathcal{M}$ with separable predual, if the norm $\mathrm{\Phi }(\cdot )$ satisfies a natural restriction introduced in (1.1), then irreducible operators are $\mathrm{\Phi }(\cdot )$-norm dense in $\mathcal{M}$. In particular, the operator norm $\mathrm{\parallel }\cdot \mathrm{\parallel }$ and the $\max{\Vert\cdot\Vert, \Vert\cdot\Vert_p}$-norm(foreach$p>1$) naturally satisfy the condition in (1.1), where $\tau$ is a faithful, normal, semifinite, tracial weight and $\mathrm{\parallel }x{\mathrm{\parallel }}_p=\tau (\mathrm{\mid }x{\mathrm{\mid }}^p{)}^{1\mathrm{/}p}$ 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 $\mathrm{\parallel }\cdot \mathrm{\parallel }$-dominating, unitarily invariant norm $\mathrm{\Phi }(\cdot )$, 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 $\mathrm{\Phi }(\cdot )$-norm perturbation, where the $\mathrm{\Phi }(\cdot )$-norm isn’t restricted by (1.1). Particularly, the $\mathrm{\Phi }(\cdot )$-norm can be the $\max \{\mathrm{\parallel }\cdot \mathrm{\parallel },\mathrm{\parallel }\cdot {\mathrm{\parallel }}_1\}$-norm.