Spectrally Bounded Operators on Finite von Neumann Algebras

这个项目解决了“重建”的问题，也被称为“逆问题”。从系统的给定数据集，以唯一的方式重建系统的数学模型。更明确地说，在微观物理学（如量子力学）中，数学模型是希尔伯特空间上的C*-代数算子，其中包含系统的可能状态。系统的可观测量只能通过测量而不能直接获取;在数学术语中，这些是（自伴）算子的本征值。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逆问题：给定两个具有相同数据集的C*-代数，它们必须同构（即，“相同的”）关于基本的代数结构。后约旦，冯诺依曼和维格纳（1934年）这是结构的约旦代数。该项目旨在解决重要类别的C*-代数的这个问题，即所谓的近似有限维C*-代数和有限冯诺依曼因子，它们与物理学最相关，因为它们分别由有限系统构建并携带忠实的有限迹。

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.