Nets of standard subspaces on causal symmetric spaces

本文利用有限维李群的反酉表示探讨了复希尔伯特空间H的标准子空间的空间Stand(H)的几何特征。这就引出了齐次空间，它具有有序膨胀空间的结构，是对Cayley型有序对称空间的推广。主要目标是理解这些空间的结构。复希尔伯特空间H中的封闭实子空间V，如果V与i V平凡相交，且它们的和在H中是稠密的，则称为标准子空间。标准子空间的空间Stand(H)带有三个明显的结构：集合包含的有序性、辛补的对偶性和由反酉算子给出的对称性。此外，还有与每个标准子空间v相关的模算子定义的对称性。因此，我们获得了一个更简单，但相当忠实的框架，用于冯·诺伊曼代数模理论基础的结构。因此，本项目特别受到量子场论（QFT）局部观测网络理论中最近发展的模块化定位技术的推动。有限维李群的反酉表示允许我们通过研究Stand(H)中有限维李群G的相应轨道来探索Stand(H)的几何。这些轨道具有相当丰富的几何结构，编码在有序膨胀空间的概念中。这个项目的主要目的是了解这些空间的结构。由于这在一般情况下是没有希望的，我们将重点放在单调膨胀测地线产生的空间上。根据Borchers和Wiesbrock的结果，这个假设在QFT的背景下是非常自然的，因为它匹配某些单参数群的正谱条件。李代数水平上的具体目标现在涉及具有相交不变凸锥的非阿贝尔二维子代数。这是可以管理的，因为由点不变凸锥生成的李代数的结构在大约20年前就已经很清楚了。在几何层面上，将有序对称空间理论推广到膨胀空间。一个核心问题是确定李群G和由G自然构造的标准子空间的一个给定的反酉表示，G的子半群由V不变的所有元素组成。

In this project we explore the geometric features of the space Stand(H) of standard subspaces of a complex Hilbert space H by antiunitary representations of finite-dimensional Lie groups. This leads us to homogeneous spaces which carry the structure of an ordered dilation spaces, a generalization of ordered symmetric spaces of Cayley type. The main goal is to understand the structure of these spaces. A closed real subspace V of a complex Hilbert space H is called standard if V intersects i V trivially and their sum is dense in H. The space Stand(H) of standard subspaces carries three obvious structures: the order by set inclusion, the duality by symplectic complements, and symmetries given by antiunitary operators. In addition, there are symmetries defined by the modular operators associated to each standard subspace V. We thus obtain a simpler, but rather faithful, framework for the structures underlying the modular theory of von Neumann algebras. Accordingly, this project is particularly motivated by the recently evolving technique of modular localization in the theory of nets of local observables in Quantum Field Theory (QFT). Antiunitary representations of finite dimensional Lie groups permit us to probe the geometry of Stand(H) by studying corresponding orbits of finite dimensional Lie groups G in Stand(H). These orbits carry rather rich geometric structure, encoded in the concept of an ordered dilation space. The main objective of this project to understand the structure of these spaces. As this is hopeless in this generality, we focus on spaces generated by monotone dilation geodesics. According to results by Borchers and Wiesbrock, this assumption is very natural in the context of QFT because it matches the positive spectrum condition on certain one-parameter groups. The concrete objectives on the Lie algebra level now concern non-abelian 2-dimensional subalgebras with intersecting invariant convex cones. This is manageable because the structure of Lie algebras generated by pointed invariant convex cones is known rather well since about 20 years. On the geometric level, the new results concern extensions of the theory of ordered symmetric spaces to dilation spaces. A core problem is to determine for a given antitunitary representation of a Lie group G and standard subspace constructed naturally from G, the subsemigroup of G, consisting of all elements leaving V invariant.