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 \({\mathcal {B}}({\mathcal {H}})\).

Halmos 在 Hal (Michigan Mathematical Journal, 15, 215–223 (1968)) 证明的一个有趣的结果是不可约算子的集合在\({\mathcal {B}}({\mathcal {H}})\ )在希尔伯特-施密特近似的意义上。在具有可分离预对数的冯诺依曼代数\({\mathcal {M}}\) 中，算子\(a\in {\mathcal {M}}\)被称为在 \({\mathcal {M}}\) 中不可约，如果\(W^*(a)\)是\({\mathcal {M}}\)的不可约子因子，即\(W^*(a) )'\cap {\mathcal {M}}={{\mathbb {C}}} \cdot I\) . 让\(\Phi (\cdot )\)成为\(\Vert \cdot \Vert \)-支配的、单一不变的范数（见定义 2.1）。许多著名的范数都是这种类型，例如运算符范数\(\Vert \cdot \Vert \)和\(\max \{\Vert \cdot \Vert , \Vert \cdot \Vert _p\}\) -norm（对于每个\(p>1\)）。在定理 3.1 中，我们证明了在每个具有可分离预对数的半有限因子\({\mathcal {M}}\) 中，如果范数\(\Phi (\cdot )\)满足（1.1）中引入的自然条件，则不可约运算符是\(\Phi (\cdot )\) -范数密集在\({\mathcal {M}}\)。特别是，运算符范数\(\Vert \cdot \Vert \)和\(\max \{\Vert \cdot \Vert , \Vert \cdot \Vert _p\}\)-norm（对于每个\(p>1\)）自然满足（1.1）中的条件。这可以看作是哈尔 (Michigan Mathematical Journal, 15, 215–223 (1968) 中的 Halmos 定理的（更强）类似物，用半有限、具有可分离预对偶的适当无限因子开发的不同技术证明。这是很自然的）问定理 3.1 是否需要 (1.1) 中的条件. 我们证明 (1.1) 中的条件对于正规算子不是必需的. 在定理 4.1 中,对于每个 \(\Vert \cdot \Vert \) -支配, 幺正不变范数 \(\Phi (\cdot )\)，我们开发了另一种方法来证明 \({\mathcal {M}}\) 是 \({\mathcal {M}}\) 中的不可约算子和任意小的 \(\Phi (\cdot )\) -范数扰动的总和。特别地，\（\披（\ CDOT）\）范数可以是跟踪类规范\（\ Vert的\ CDOT \ Vert的_1 \）上在该组迹类算子的\（{\ mathcal {B}}（ {\mathcal {H}})\)。