When a homogeneous convex cone is given, a natural partial order is introduced in the cone. We shall show that a homogeneous convex cone is a symmetric cone if and only if Vinberg´s $\ast$
-map and its inverse reverse the order. Actually our theorem is formulated in terms of the family of pseudoinverse maps including the $\ast$
-map, and states that the above order-reversing property is typical of the $\ast$
-map of a symmetric cone which coincides with the inverse map of the Jordan algebra associated with the symmetric cone.
@article{1225894035,
author = {KAI, Chifune},
title = {A characterization of symmetric cones by an order-reversing property of the pseudoinverse maps},
journal = {J. Math. Soc. Japan},
volume = {60},
number = {1},
year = {2008},
pages = { 1107-1134},
language = {en},
url = {http://dml.mathdoc.fr/item/1225894035}
}
KAI, Chifune. A characterization of symmetric cones by an order-reversing property of the pseudoinverse maps. J. Math. Soc. Japan, Tome 60 (2008) no. 1, pp. 1107-1134. http://gdmltest.u-ga.fr/item/1225894035/