Advertisement

Inventiones mathematicae

, Volume 115, Issue 1, pp 463–482 | Cite as

Sharp uniform convexity and smoothness inequalities for trace norms

  • Keith Ball
  • Eric A. Carlen
  • Elliott H. Lieb
Article

Summary

We prove several sharp inequalities specifying the uniform convexity and uniform smoothness properties of the Schatten trace idealsC p , which are the analogs of the Lebesgue spacesL p in non-commutative integration. The inequalities are all precise analogs of results which had been known inL p , but were only known inC p for special values ofp. In the course of our treatment of uniform convexity and smoothness inequalities forC p we obtain new and simple proofs of the known inequalities forL p .

Keywords

Simple Proof Trace Norm Uniform Convexity Smoothness Property Sharp Inequality 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [ArYa] Araki, H., Yamagami, S.: An inequality for the Hilbert-Schmidt norm. Commun. Math. Phys.81, 89–96 (1981)Google Scholar
  2. [BP] Ball, K., Pisier, G.: Unpublished result; private communication.Google Scholar
  3. [Bo] Boas, R.P.: Some uniformly convex spaces. Bull. Am. Math. Soc.46, 304–311 (1940)Google Scholar
  4. [C] Clarkson, J.A.: Uniformly convex spaces. Trans. Am. Math. Soc.40, 396–414 (1936)Google Scholar
  5. [CL] Carlen, E., Lieb, E.: Optimal hypercontractivity for fermi fields and related non-commutative integration inequalities. Commun. Math. Phys. “155, 27–46 (1993); for a slightly different presentation, see: Optimal two-uniform convexity and fermion hypercontractivity. In: Araki, H., Ito, K.R., Kishimoto, A., Ojima, I. (eds.) Quantum and non-commutative analysis. London New York. Kluwer (in press)Google Scholar
  6. [D] Day, M.: Uniform convexity in factor and conjugate spaces. Ann. Math.45, 375–385 (1944)Google Scholar
  7. [Di] Dixmier, J.: Formes linéaires sur un anneau d'opérateurs. Bull. Soc. Math. Fr.81, 222–245 (1953)Google Scholar
  8. [F] Figiel, T.: On the moduli of convexity and smoothness. Studia Math.56, 121–155 (1976)Google Scholar
  9. [FJ] Figiel, T., Johnson, S.B.: A uniformly convex Banach space which contains noC p. Compos. Math.29, 179–190 (1974)Google Scholar
  10. [Gr] Gross, L.: Logarithmic Sobolev inequalities. Am. J. Math.97, 1061–1083 (1975)Google Scholar
  11. [H] Hanner, O.: On the uniform convexity ofL p andl p. Ark. Math.3, 239–244 (1956)Google Scholar
  12. [Kö] Köthe, G.: Topologische lineare Räume, Die Grundlehren der mathematischen Wissen schaften in Einzeldarstellungen, Bd. 107, Springer Berlin Heidelberg New York: 1960Google Scholar
  13. [L] Lindenstrauss, J.: On the modulus of smoothness and divergent series in Banach spaces. Mich. Math. J.10, 241–252 (1963)Google Scholar
  14. [P] Pisier, G.: The volume of convex bodies and Banach space geometry. Cambridge: Cambridge University Press, 1989Google Scholar
  15. [Ru] Ruskai, M.B.: Inequalities for traces on Von Neumann algebras. Commun. Math. Phys.26, 280–289 (1972)Google Scholar
  16. [Se] Segal, I.E.: A non-commutative extension of abstract integration. Ann. Math.57, 401–457 (1953)Google Scholar
  17. [Si] Simon, B.: Trace ideals and their applications. (See p. 22) Cambridge: Cambridge University Press, 1979Google Scholar
  18. [TJ] Tomczak-Jaegermann, N.: The moduli of smoothness and convexity and Rademacher averages of trace classesS p(1≦p<∞). Studia Math.50, 163–182 (1974)Google Scholar

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • Keith Ball
    • 1
  • Eric A. Carlen
    • 2
  • Elliott H. Lieb
    • 3
  1. 1.Department of MathematicsTexas A&M UniversityCollege StationUSA
  2. 2.Georgia Institute of TechnologySchool of MathematicsAtlantaUSA
  3. 3.Departments of Mathematics and PhysicsPrinceton UniversityPrincetonUSA

Personalised recommendations