## Journal of the Mathematical Society of Japan

### A characterization of symmetric cones by an order-reversing property of the pseudoinverse maps

Chifune KAI

#### Abstract

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 information

Source
J. Math. Soc. Japan, Volume 60, Number 4 (2008), 1107-1134.

Dates
First available in Project Euclid: 5 November 2008

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1225894035

Digital Object Identifier
doi:10.2969/jmsj/06041107

Mathematical Reviews number (MathSciNet)
MR2467872

Zentralblatt MATH identifier
1167.32015

#### Citation

KAI, Chifune. A characterization of symmetric cones by an order-reversing property of the pseudoinverse maps. J. Math. Soc. Japan 60 (2008), no. 4, 1107--1134. doi:10.2969/jmsj/06041107. https://projecteuclid.org/euclid.jmsj/1225894035

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