Spectrally Bounded Operators on Finite von Neumann Algebras

这个项目解决了“重建”的问题，也称为“逆问题”。根据系统的一组给定数据，系统的数学模型将以独特的方式重建。更明确地说，在量子力学等微观物理学中，数学模型是希尔伯特空间上的C*-算子代数之一，其中包含系统的可能状态。系统的可观量只能通过测量来恢复，不能直接获得；用数学术语来说，这些是(自伴)算符的特征值。一旦选择了模型(C*-代数)，数据集(特征值)就固定了。我们研究了数据集在多大程度上预先决定模型选择的反问题：给定两个具有相同数据集的C*-代数，它们关于基本代数结构是否必须同构(即‘相同’)。在Jordan，von Neumann和Wigner(1934)之后，这是Jordan代数的结构。这个项目旨在为重要的C*-代数解决这个问题，即所谓的近似有限维C*-代数和有限von Neumann因子，它们与物理最相关，因为它们是从有限系统建立的，并且分别具有忠实的有限迹。

This project addresses the question of 'reconstruction', also called an 'inverse problem'. From a given set of data of a system the mathematical model for the system is to be reconstructed in a unique way. More explicitly, in microscopic physics such as quantum mechanics the mathematical model is the one of a C*-algebra of operators on a Hilbert space, which contains the possible states of the system. The observables of the system can only be retrieved through measurements, not directly; in mathematical terms these are the eigenvalues of the (selfadjoint) operators. Once the model (the C*-algebra) is chosen, the set of data (eigenvalues) is fixed. We study the inverse question to what extent the set of data pre-determines the choice of the model: given two C*-algebras with the same data set, do they have to be isomorphic (i.e., 'the same') with regard to the essential algebraic structure. After Jordan, von Neumann and Wigner (1934) this is the structure of a Jordan algebra. This projects intends to solve this problem for important classes of C*-algebras, the so-called approximately finite dimensional C*-algebras and the finite von Neumann factors, which are most relevant for physics as they are built from finite systems and carry a faithful finite trace, respectively.