Order antimorphisms of finite-dimensional cones
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Order antimorphisms of finite-dimensional cones Cormac Walsh1
© Springer Nature Switzerland AG 2020
Abstract We show that an order antimorphism on a finite-dimensional cone having no onedimensional factors is homogeneous of degree − 1. A consequence is that the only finite-dimensional cones admitting an order antimorphism are the symmetric cones. Mathematics Subject Classification 46A40
1 Introduction Given an order structure, it is natural to study its isomorphisms and antimorphisms. Recall that the former are the bijective maps that preserve the order in both directions, and the latter are those that reverse the order in both directions. Here, we study the antimorphisms of finite-dimensional ordered vector spaces. The natural domain in this setting is the open cone of positive elements. We are interested in knowing when a particular form of rigidity holds, namely, when all antimorphisms are necessarily antihomogeneous, that is, satisfy φ(λx) = λ−1 φ(x), for all elements x of the cone, and λ > 0. We will see that this is the case for certain ordered vector spaces, for instance, the space of Hermitian matrices of fixed dimension ordered by the cone of those that are positive definite. In this space, the map A → A−1 is an example of an antihomogeneous antimorphism. On the other hand, there are spaces for which the rigidity phenomenon does not occur. An obvious example is the one-dimensional space R with the cone (0, ∞). Here, any monotonically decreasing homeomorphism of the cone is an order antimorphism, but clearly there are many such maps that are not antihomogeneous. A more general example is the product of a one-dimensional cone with any cone that admits an antimorphism—simply have this antimorphism operate on the cone factor and have a non-antihomogeneous antimorphism operate on the one-dimensional factor.
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Cormac Walsh [email protected] Inria and CMAP, École Polytechnique, CNRS, Université Paris-Saclay, 91128 Palaiseau Cedex, France 0123456789().: V,-vol
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C. Walsh
Our main result is that, when we exclude the existence of one-dimensional factors, we get the rigidity we desire. Recall that a proper cone is one that contains no complete lines. Theorem 1.1 Let φ : C → C be an order antimorphism between two proper open convex cones in a finite-dimensional linear space. If C has no one-dimensional factors, then φ is antihomogeneous. In [17] it was shown that there can be an antihomogeneous order-antimorphism between two finite-dimensional cones only if both cones are symmetric, that is, homogeneous and self-dual. Combining this with the theorem above, we get the following. Corollary 1.2 Let φ : C → C be an order antimorphism between two proper open convex cones in a finite-dimensional linear space. Then C and C are symmetric cones. The technique in [17] was to consider the Funk metric on the cone, which is a non-symmetric metric defined using the order and homogeneity structures. Each antihomogeneous antimorphism on a cone reverses this metric [
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