Let $\mathcal{M}$ be a von Neumann algebra, $\mathcal{I}$ a weak-operator dense ideal in $\mathcal{M}$, and Φ a unitarily invariant ∥ · ∥-dominating norm on $\mathcal{I}$. In this talk, we provide a necessary and sufficient condition on  Φ such that every operator in  $\mathcal{M}$ can be expressed as the sum of an operator in $\mathcal{M}$ with disconnected spectrum and an operator in $\mathcal{I}$ whose Φ-norm is arbitrarily small. Similarly, if $\mathcal{A}$ is a unital $C^∗$-algebra of real rank zero with dimension greater than one and $\mathcal{I}$ is an essential ideal in $\mathcal{A}$, then every element in $\mathcal{A}$ can be written as the sum of an operator in $\mathcal{A}$ with disconnected spectrum and an operator in $\mathcal{I}$ whose norm is arbitrarily small.

石瑞，大连理工大学数学科学学院教授，博士生导师，主要从事泛函分析中算子理论及算子代数相关的研究工作；近些年围绕算子代数中的算子结构理论、约化理论、分类理论，以及算子代数的表示理论等取得了一些相关研究成果；代表性科研论文发表于《Adv. Math.》、《Math. A》、《J. Funct. Anal.》、《J. Noncommut. Geom.》、《Integral Equations Operator Theory》、《Proc. Amer. Math. Soc. 》等学术期刊。