Osaka Journal of Mathematics

Metric convexity of symmetric cones

Abstract

In this paper we 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. As a special case, we 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 Neeb [22].

Article information

Source
Osaka J. Math., Volume 44, Number 4 (2007), 795-816.

Dates
First available in Project Euclid: 7 January 2008

https://projecteuclid.org/euclid.ojm/1199719405

Mathematical Reviews number (MathSciNet)
MR2383810

Zentralblatt MATH identifier
1135.53014

Citation

Lawson, Jimmie; Lim, Yongdo. Metric convexity of symmetric cones. Osaka J. Math. 44 (2007), no. 4, 795--816. https://projecteuclid.org/euclid.ojm/1199719405

References

• E.M. Alfsen, F.W. Shultz and E. Størmer: A Gelfand-Neumark theorem for Jordan algebras, Advances in Math. 28 (1978), 11--56.
• T. Ando: Topics on Operator Inequalities, Lecture Notes Hokkaido Univ., Sapporo, 1978.
• E. Andruchow, G. Corach and D. Stojanoff: Geometrical significance of Löwner-Heinz inequality, Proc. Amer. Math. Soc. 128 (2000), 1031--1037.
• E. Andruchow, G. Corach, and D. Stojanoff: Löwner's theorem and the differential geometry of the space positive operators, preprint.
• W. Ballmann: Lectures on Spaces of Nonpositive Curvature, DMV Seminar 25, Birkhäuser, Basel, 1995.
• R. Bhatia: On the exponential metric increasing property, Linear Algebra Appl. 375 (2003), 211--220.
• R. Braun, W. Kaup and H. Upmeier: A holomorphic characterization of Jordan $C^*$-algebras, Math. Z. 161 (1978), 277--290.
• M.R. Bridson and A. Haefliger: Metric Spaces of Non-Positive Curvature, Grundlehren der Math. Wissenschaft 319, Springer, Berlin, 1999.
• G. Corach, H. Porta and L. Recht: Convexity of the geodesic distance on spaces of positive operators, Illinois J. Math. 38 (1994), 87--94.
• P. Eberlein: Geometry of Non-Positively Curved Manifolds, Chicago Lectures in Math., U. Chicago Press, 1996.
• T. Furuta: $A\geq B\geq 0$ assures $(B^rA^pB^r)^1/q\geq B^(p+2r)/q$ for $r\geq 0$, $p\geq 0$, $q\geq 1$ with $(1+2r)q\geq p+2r$, Proc. Amer. Math. Soc. 101 (1987), 85--88.
• H. Hanche-Olsen and E. Størmer: Jordan Operator Algebras, Monographs and Studies in Math. 21, Pitman, Boston, 1984.
• S. Lang: Fundamentals of Differential Geometry, Graduate Texts in Math. 191, Springer, Heidelberg, 1999.
• J. Lawson and Y. Lim: Symmetric sets with midpoints and algebraically equivalent theories, Results Math. 46 (2004), 37--56.
• J. Lawson and Y. Lim: Means on dyadic symmetric sets and polar decompositions, Abh. Math. Sem. Univ. Hamburg 74 (2004), 135--150.
• J. Lawson and Y. Lim: Symmetric spaces with convex metrics, to appear, Forum Math.
• J. Lawson and Y. Lim: Solving symmetric matrix word equations via symmetric space machinery, Linear Algebra Appl. 414 (2006), 560--569.
• Y. Lim: Finsler metrics on symmetric cones, Math. Ann. 316 (2000), 379--389.
• Y. Lim: Geometric means on symmetric cones, Arch. Math. (Basel) 75 (2000), 39--45.
• Y. Lim: Applications of geometric means on symmetric cones, Math. Ann. 319 (2001), 457\nobreakdash--468.
• O. Loos: Symmetric Spaces, I: General Theory, W.A. Benjamin, Inc., New York-Amsterdam, 1969.
• K.-H. Neeb: A Cartan-Hadamard theorem for Banach-Finsler manifolds, Geom. Dedicata 95 (2002), 115--156.
• R.D. Nussbaum: Hilbert's Projective Metric and Iterated Nonlinear Maps, Memoirs of Amer. Math. Soc. 391, 1988.
• R.D. Nussbaum: Finsler structures for the part metric and Hilbert's projective metric and applications to ordinary differential equations, Differential Integral Equations 7 (1994), 1649--1707.
• H. Olsen and E. Størmer: Jordan Operator Algebras, Monographs Stud. Math. 21, Pitman, London, 1984.
• S. Shirali and J.W.M. Ford: Symmetry in complex involutory Banach algebras II, Duke Math. J. 37 (1970), 275--280.
• A.C. Thompson: On certain contraction mappings in a partially ordered vector space, Proc. Amer. Math. soc. 14 (1963), 438--443.
• K. Tanahashi and A. Uchiyama: The Furuta inequality in Banach $\ast$-algebras, Proc. Amer. Math. Soc. 128 (2000), 1691--1695.
• H. Upmeier: Symmetric Banach Manifolds and Jordan $C^\ast$-algebras, North-Holland Mathematics Studies 104 North-Holland, Amsterdam, 1985.
• J.D.M. Wright: Jordan $C\sp*$-algebras, Michigan Math. J. 24 (1977), 291--302.