Spectrally Bounded Operators on Finite von Neumann Algebras

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.