Metric convexity of symmetric cones. (English) Zbl 1135.53014

Lawson, Jimmie; Lim, Yongdo

Osaka J. Math. 44, No. 4, 795-816 (2007).

Using an algebraic construction of symmetric sets, the authors introduce a general notion of a symmetric cone, valid for the finite and infinite dimensional case, and prove that one can deduce the seminegative curvature of the Thompson part metric in this general setting, along with standard inequalities familiar from operator theory. A special case is the symmetric cone arising as the set of invertible squares of a Jordan-Banach algebra (JB-algebra). The authors prove that every symmetric cone from a JB-algebra satisfies a certain convexity property for the Thompson part metric: the distance function between points evolving in time on two geodesics is a convex function. This provides an affirmative answer to a question of K.-H. Neeb [Geom. Dedicata 95, 115–156 (2002; Zbl 1027.58003)].

Reviewer: Sergei Platonov (Petrozavodsk)

MSC:

53B40	Local differential geometry of Finsler spaces and generalizations (areal metrics)
53C35	Differential geometry of symmetric spaces
46H70	Nonassociative topological algebras
46B20	Geometry and structure of normed linear spaces

Keywords:

symmetric cones; metric convexity; symmetric sets; Jordan-Banach algebra

Citations:

Zbl 1027.58003

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