On the geometry of standard subspaces

TL;DR

This paper explores the geometric structure of standard subspaces in complex Hilbert spaces, revealing their reflection and dilation space properties, and connecting them to antiunitary representations and Jordan algebras.

Contribution

It introduces the reflection and dilation space structures of standard subspaces and links these to antiunitary representations and Jordan algebra embeddings.

Findings

Modular conjugations define a reflection space structure.

Modular automorphism groups extend this to a dilation space.

Ordered symmetric spaces from Jordan algebras embed into standard subspaces.

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$_{gr}(R^x,G)$ 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.