Contractive Projections, Quantum Mechanics and Operator Spaces

University Of California-Irvine, Irvine CA

Investigators

Abstract

This research plan consists of two themes. The first is concerned with investigations into the structure theory of injective-like objects in the category of operator spaces, and involves the theory of contractive projections on real or complex C*-algebras. The second is a study of linear and nonlinear representations of the Lorentz group with an eye towards finding physically meaningful models in mathematical physics. The link connecting these two objectives is described in the following paragraph. Since the pioneering work of Jordan, von Neumann and Wigner in the 1930s, Segal in the 1940s and 50s, and Alfsen-Shultz-Stormer in the 1970s, Jordan algebras, and more recently Jordan triples, have been used as a model for the study of the state spaces of quantum mechanics. On the other hand, contractive projections have played a key role in the structure theory of Banach Jordan triple systems, and those which are completely contractive give rise to injectives in the category of operator spaces (also known as quantized Banach spaces). The PIs plan to study the fine structure of contractive projections on operator spaces in order to gain a better understanding of their role in operator space theory, as well as their role, together with Jordan structures, in quantum mechanics and relativity.