Normed ideal perturbation of irreducible operators in semifinite von Neumann factors

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 $W^{\ast }(a)$ is an irreducible subfactor of $\mathcal{M}$, i.e., $W^{\ast }(a{)}^{\prime }\cap \mathcal{M}=\mathbb{C}\cdot I$. In this paper, let $\hat{I}¦(\cdot )$ 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 $\hat{I}¦(\cdot )$ satisfies a natural restriction introduced in (1.1), then irreducible operators are $\hat{I}¦(\cdot )$-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 $\ddot{I}„$ is a faithful, normal, semifinite, tracial weight and $\Vert x{\Vert }_p=\ddot{I}„(|x{|}^p{)}^{1/p}$ for all $x\in \mathcal{M}\cap L^p(\mathcal{M},\ddot{I}„)$ (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 $\hat{I}¦(\cdot )$, 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 $\hat{I}¦(\cdot )$-norm perturbation, where the $\hat{I}¦(\cdot )$-norm isn't restricted by (1.1). Particularly, the $\hat{I}¦(\cdot )$-norm can be the $\max \{\Vert \cdot \Vert ,\Vert \cdot {\Vert }_1\}$-norm.

25 pages