Nets of standard subspaces on causal symmetric spaces

在这个项目中，我们探讨了空间Stand（H）的几何特征的标准子空间的复希尔伯特空间H有限维李群的反对酉表示。这使我们的齐性空间进行结构的一个有序膨胀空间，一个推广的有序对称空间的凯莱型。主要目的是了解这些空间的结构。复Hilbert空间H的一个闭真实的子空间V称为标准的，如果V与iV平凡相交且它们的和在H中稠密.标准子空间的空间Stand（H）具有三种明显的结构：集合包含的序、辛补的对偶和反酉算子的对称。此外，有对称性定义的模块化运营商与每个标准子空间V，因此，我们得到了一个更简单的，但相当忠实的，框架的结构基础的模块化理论的冯诺依曼代数。因此，这个项目是特别由最近不断发展的技术模块本地化在量子场论（QFT）的局部观测网络理论的动机。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。这些轨道带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。由于这是无望的，在这种普遍性，我们专注于单调膨胀测地线所产生的空间。根据Borchers和Wiesbrock的结果，这个假设在QFT的上下文中是非常自然的，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关注的非交换2维子代数相交不变凸锥。这是可管理的，因为由尖不变凸锥生成的李代数的结构在大约20年前就已经相当清楚了。在几何水平上，新的结果涉及扩展的理论有序对称空间的膨胀空间。一个核心问题是确定一个给定的李群G的反酉表示和标准子空间自然地从G，G的子半群，由所有元素离开V不变。

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.