Contractive Projections, Quantum Mechanics and Operator Spaces

本研究计划由两个主题组成。第一部分是关于算子空间范畴内射类对象的结构理论的研究，涉及实或复C*-代数上的压缩投影理论。第二个是研究洛伦兹群的线性和非线性表示，着眼于在数学物理中寻找有物理意义的模型。自从乔丹，冯·诺伊曼和维格纳在20世纪30年代，西格尔在40年代和50年代，阿尔夫森-舒尔茨-斯托默在70年代的开创性工作以来，乔丹止痛剂，以及最近的乔丹三元组，一直被用作研究量子力学状态空间的模型。另一方面，压缩投影在Banach Jordan三元系的结构理论中起着关键作用，而完全压缩的投影在算子空间范畴(也称为量子化的Banach空间)中产生内射。PI计划研究算子空间上压缩投影的精细结构，以便更好地理解它们在算子空间理论中的作用，以及它们与Jordan结构一起在量子力学和相对论中的作用。

This research plan consists of two themes. The first is concernedwith investigations into the structure theory of injective-like objects inthe category of operator spaces, and involves the theory ofcontractive projections on real or complex C*-algebras. The second isa study of linear and nonlinear representations of the Lorentz group withan eye towards finding physically meaningful models in mathematicalphysics. The link connecting these two objectives is described in thefollowing 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, Jordanalgebras, and more recently Jordan triples, have been used as a model forthe study of the state spaces of quantum mechanics. On the other hand,contractive projections have played a key role in the structure theory ofBanach Jordan triple systems, and those which are completely contractivegive rise to injectives in the category of operator spaces (also known asquantized Banach spaces). The PIs plan to study the fine structure ofcontractive projections on operator spaces in order to gain a betterunderstanding of their role in operator space theory, as well as theirrole, together with Jordan structures, in quantum mechanics andrelativity.