Nets of standard subspaces on causal symmetric spaces

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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.