On the geometry of standard subspaces
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Publication:4961930
DOI10.1090/CONM/714/14330zbMATH Open1406.22015arXiv1707.05506OpenAlexW2964215694MaRDI QIDQ4961930
Author name not available (Why is that?)
Publication date: 29 October 2018
Published in: (Search for Journal in Brave)
Abstract: A closed real subspace V of a complex Hilbert space H is called standard if V intersects iV trivially and and V + i V is dense in H. In this note we study several aspects of the geometry of the space Stand(H) of standard subspaces. In particular, we show that modular conjugations define the structure of a reflection space and that the modular automorphism groups extend this to the structure of a dilation space. Every antiunitary representation of a graded Lie group G leads to a morphism of dilation spaces Hom to Stand(H). Here dilation invariant geodesics (with respect to the reflection space structure) correspond to antiunitary representations U of Aff(R) and they are decreasing if and only if U is a positive energy representation. We also show that the ordered symmetric spaces corresponding to euclidean Jordan algebras have natural order embeddings into Stand(H) obtained from any antiunitary positive energy representations of the conformal group.
Full work available at URL: https://arxiv.org/abs/1707.05506
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