Normed Ideal Perturbation of Irreducible Operators in Semifinite Von Neumann Factors

Abstract

An interesting result proved by Halmos in Hal (Michigan Mathematical Journal, 15, 215–223 (1968) is 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., $W^*(a)'\cap {\mathcal {M}}={{\mathbb {C}}} \cdot I$. Let $\Phi (\cdot )$ be a $\Vert \cdot \Vert$-dominating, unitarily invariant norm (see Definition 2.1). Many well-known norms are of this type such as the operator norm $\Vert \cdot \Vert$ and the $\max \{\Vert \cdot \Vert , \Vert \cdot \Vert _p\}$-norm (for each $p>1$). In Theorem 3.1, we prove that in every semifinite factor ${\mathcal {M}}$ with separable predual, if the norm $\Phi (\cdot )$ satisfies a natural condition 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). This can be viewed as a (stronger) analogue of a theorem of Halmos in Hal (Michigan Mathematical Journal, 15, 215–223 (1968), proved with different techniques developed in semifinite, properly infinite factors with separable predual. It is natural to ask whether the condition in (1.1) is necessary for Theorem 3.1. We prove that the condition in (1.1) is not necessary for normal operators. In Theorem 4.1, 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. Particularly, the $\Phi (\cdot )$-norm can be the trace class norm $\Vert \cdot \Vert _1$ on the set of trace class operators in ${\mathcal {B}}({\mathcal {H}})$.