冯·诺依曼代数的可测算子的性质

Properties of Measurable Operators Associated with a von Neumann Algebra

本文研究了冯·诺依曼代数的可测算子的基本性质,定义了阶梯算子,证明了任意一个正可测算子可以由阶梯算子在定义域内按照强算子拓扑逼近,从而证明了任意一个可测算子可以由投影在定义域内按照强算子拓扑逼近.此外,还讨论了可测算子与有界算子的复合算子的可测性.

In this article, some properties of measurable operators associated with a von Neumann algebra are considered. The concept of step operator is defined and it is proved that any positive measurable operator can be strongly approximated by some step operators on its domain, which means that any positive measurable operator can be strongly approximated by some projections on its domain. In addition, the measurability of composition operator of measurable operator and bounded operator is discussed.