循环向量和分离向量的特征化及其在模论反问题中的应用I。有限因素
Chracterization of Cyclic and Separating Vectors and Application to an Inverse Problem in Modular Theory I. Finite Factors
摘要：
本文研究了有限因子情况下冯诺依曼代数模理论中的一个反问题。首先，我们给出了一个特征的循环和分离向量的有限因素的运营商与这个向量，并隶属于该因素。此外，我们展示了该算子如何生成给定循环和分离向量的模块化对象，推广了Kadison和Ringrose的思想。借助这些相当技术性的结果，我们表明在一个适当的条件下，这总是满足有限型I因子，存在另一类简单的解决方案的反问题旁边的平凡的总是存在的。最后给出了具有纯点谱的模算子的反问题解的完全分类，这在第Ⅰ类情形下是没有限制的。在随后的论文中，这些结果将推广到所有的半有限因子。
英文摘要：
In this paper we examine an inverse problem in the modular theory of von Neumann algebras in the case of finite factors. First we give a characterization of cyclic and separating vectors for finite factors in terms of operators associated with this vector and being affiliated with the factor. Further we show how this operator generates the modular objects of the given cyclic and separating vector generalizing an idea of Kadison and Ringrose. With the help of these rather technical results we show under an appropriate condition, which is always fulfilled for finite type I factors, that there exists another simple class of solutions of the inverse problem beside a trivial one which always exists. Finally we give a complete classification of the solutions of the inverse problem in the case of modular operators having pure point spectrum which is no restriction in the type I case. In a subsequent paper these results will be generalized to all semifinite factors.
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