Positive and doubly stochastic maps, and majorization in Euclidean Jordan algebras

A positive map between Euclidean Jordan algebras is a (symmetric cone) order preserving linear map. We show that the norm of such a map is attained at the unit element, thus obtaining an analog of the operator/matrix theoretic Russo Dye theorem. A doubly stochastic map between Euclidean Jordan algebras is a positive, unital, and trace preserving map. We relate such maps to Jordan algebra automorphisms and majorization. (C) 2016 Elsevier Inc. All rights reserved.