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On the geometry of standard subspaces
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
Discounted Hamilton-Jacobi Equations on Networks and Asymptotic Analysis
We study discounted Hamilton Jacobi equations on networks, without putting
any restriction on their geometry. Assuming the Hamiltonians continuous and
coercive, we establish a comparison principle and provide representation
formulae for solutions. We follow the approach introduced in 11, namely we
associate to the differential problem on the network, a discrete functional
equation on an abstract underlying graph. We perform some qualitative analysis
and single out a distinguished subset of vertices, called lambda Aubry set,
which shares some properties of the Aubry set for Eikonal equations on compact
manifolds. We finally study the asymptotic behavior of solutions and lambda
Aubry sets as the discount factor lambda becomes infinitesimal.Comment: Corrected typos in the titl
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