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Section: New Results

Non-linear Perron-Frobenius theory, nonexpansive mappings and metric geometry

Order reversing maps on cones

Participant : Cormac Walsh.

We have been studying non-linear operators on open cones, particularly ones that preserve or reverse the order structure associated to the cone. A bijective map that preserves the order in both directions is called an order isomorphism. Those that reverse the order in both directions are order antimorphisms. These are closely related to the isometries of the Hilbert and Thompson metrics on the cone.

Previously, we have shown  [118] that if there exists an antimorphism on an open cone that is homogeneous of degree $-1$, then the cone must be a symmetric cone, that is, have a transitive group of linear automorphisms and be self-dual.

The technique was to consider the Funk metric on the cone, which is a non-symmetric metric defined using the order and homogeneity structures. Each antimorphism on a cone that is homogeneous of degree $-1$ reverses this metric, and so interchanges the two horofunction boundaries of the metric, the one in the forward direction and the one in the backward direction. By studying these boundaries we obtained the result.

More recently, we have shown [45] that the homogeneity assumption is not actually necessary: every antimorphism on a cone is automatically homogeneous of degree $-1$.

Without the homogeneity assumption, the metric techniques do not work. Instead, it was necessary to study how the map acts on line segments parallel to extreme rays of the cone. This is similar to the what was done by Rothaus, Noll and Schäffer, and Artstein-Avidan and Slomka in their work on order isomorphisms. This means that our proof is essentially finite dimensional. Indeed, there are many interesting cones in infinite dimension that have few or no extreme rays.

In infinite dimension, it is natural to consider order unit spaces as a generalisation of cones, and JB-algebras as a generalisation of symmetric cones. Lemmens, Roelands, and Wortel have asked whether the existence of an order antimorphism that is homogeneous of degree $-1$ on the cone of an order-unit space implies that the space is a JB-algebra? The result above suggests further that one might even be able to drop the homogeneity assumption.

The set of minimal upper bounds of two matrices in the Loewner order

Participant : Nikolas Stott.

A classical theorem of Kadison shows that the space of symmetric matrices equipped with the Loewner order is an anti-lattice, meaning that two matrices have a least upper bound if and only if they are comparable. In  [115], we refined this theorem by characterizing the set of minimal upper bounds: we showed that it is homeomorphic to the quotient space $O\left(p\right)\setminus O(p,q)/O\left(q\right)$, where $O(p,q)$ denotes the orthogonal group associated to the quadratic form with signature $(p,q)$, and $O\left(p\right)$ denotes the standard $p$th orthogonal group.

Checking the strict positivity of Kraus maps is NP-hard

Participant : Stéphane Gaubert.

In collaboration with Zheng Qu (now with HKU, Hong Kong), I studied several decision problems arising from the spectral theory of Kraus maps (trace preserving completely positive maps), acting on the cone of positive semidefinite matrices. The latter appear in quantum information. We showed that checking the irreducibility (absence of non-trivial invariant face of the cone) and primitivity properties (requiring the iterates of the map to send the cone to its interior) can be checked in polynomial time, whereas checking positivity (whether the map sends the cone to its interior) is NP-hard. In [20], we studied complexity issues related to Kraus maps, and showed in particular that checking whether a Kraus map sends the cone to its interior is NP-hard.