Contractive Projections, Quantum Mechanics and Operator Spaces

本研究计划包括两个主题。第一部分研究算子空间范畴内类内射对象的结构理论，涉及到真实的或复C ~*-代数上的压缩投影理论。第二个伊萨研究线性和非线性表示的洛伦兹群，着眼于寻找物理上有意义的模型。连接这两个目标的联系在下面的段落中描述。自20世纪30年代Jordan、von Neumann和Wigner，20世纪40年代和50年代Segal以及20世纪70年代Alfsen-Shultz-Stormer的开创性工作以来，Jordan代数以及最近的Jordan三元组一直被用作量子力学状态空间研究的模型。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，而完全压缩投影又在算子空间（也称为量子化Banach空间）范畴中产生了内射。PI计划研究算子空间上的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，以及它们与约旦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.