On some Finsler structures of symmetric cones. (English) Zbl 0990.53073
The authors study the Finsler metric defined on the domain of positive invertible elements of a \(C^*\)-algebra, which is faithfully represented in a Hilbert space. This domain is a special case of a symmetric cone from JB-algebras, the selfadjoint part of JB*-algebras. They consider only the finite-dimensional case.
The main results: 1) Determination of some Finsler metrics on the symmetric cones in the context of G. Corach, H. Porta and L. Recht [Proc. Am. Math. Soc. 115, 229-231 (1992; Zbl 0749.58010); Int. J. Math. 4, 193-202 (1993; Zbl 0809.47017)]. 2) A symmetric cone \(\Omega\) has some sort of nonpositive curvature with respect to the considered Finsler metric.
The main results: 1) Determination of some Finsler metrics on the symmetric cones in the context of G. Corach, H. Porta and L. Recht [Proc. Am. Math. Soc. 115, 229-231 (1992; Zbl 0749.58010); Int. J. Math. 4, 193-202 (1993; Zbl 0809.47017)]. 2) A symmetric cone \(\Omega\) has some sort of nonpositive curvature with respect to the considered Finsler metric.
Reviewer: R.Miron (Iaşi)
MSC:
| 53C60 | Global differential geometry of Finsler spaces and generalizations (areal metrics) |
| 53C30 | Differential geometry of homogeneous manifolds |
| 32M15 | Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects) |
| 58B20 | Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds |
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