Journal of the Mathematical Society of Japan

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

Chifune KAI

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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 -map and its inverse reverse the order. Actually our theorem is formulated in terms of the family of pseudoinverse maps including the -map, and states that the above order-reversing property is typical of the -map of a symmetric cone which coincides with the inverse map of the Jordan algebra associated with the symmetric cone.

Article information

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

First available in Project Euclid: 5 November 2008

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Zentralblatt MATH identifier

Primary: 32M15: Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras [See also 22E10, 22E40, 53C35, 57T15]
Secondary: 53C30: Homogeneous manifolds [See also 14M15, 14M17, 32M10, 57T15] 53C35: Symmetric spaces [See also 32M15, 57T15]

homogeneous convex cone symmetric cone partial order pseudoinverse map duality mapping


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.

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