Nets of standard subspaces on causal symmetric spaces

在这个项目中，我们利用有限维李群的反么正表示来研究复Hilbert空间H的标准子空间的空间站(H)的几何特征。这将我们引向具有有序扩张空间结构的齐次空间，有序扩张空间是Cayle型有序对称空间的推广。主要目标是了解这些空间的结构。复Hilbert空间H的闭实子空间V称为标准空间，如果V与V平凡相交，且它们的和在H中稠密。标准子空间的空间站(H)具有三种明显的结构：依集包含序、依辛补对偶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相关的模算子所定义的对称性。因此，我们得到了von Neumann代数模理论背后的结构的一个更简单但相当忠实的框架。因此，这个项目特别受到量子场论(QFT)中局域观测网络理论中最近发展的模块局部化技术的推动。有限维李群的反么正表示允许我们通过研究有限维李群G在林分(H)中的相应轨道来探索林分(H)的几何。这些轨道携带着相当丰富的几何结构，编码在有序膨胀空间的概念中。这个项目的主要目标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，我们主要讨论由单调伸缩测地线生成的空间。根据Borcher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上的正谱条件。李代数层次上的具体目标现在涉及到具有相交不变凸锥的非交换二维子代数。这是可控的，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。在几何水平上，新的结果涉及到序对称空间理论到扩张空间的推广。一个核心问题是确定给定的李群G的反调谐表示和由G自然构成的标准子空间，G是G的子半群，由所有保持V不变的元素组成。

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.