Annals of Functional Analysis

On the Araki–Lieb–Thirring inequality in the semifinite von Neumann algebra

Yazhou Han

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


This paper extends a recent matrix trace inequality of Bourin–Lee to semifinite von Neumann algebras. This provides a generalization of the Lieb–Thirring-type inequality in von Neumann algebras due to Kosaki. Some new inequalities, even in the matrix case, are also given for the Heinz means.

Article information

Ann. Funct. Anal., Volume 7, Number 4 (2016), 622-635.

Received: 15 December 2015
Accepted: 5 May 2016
First available in Project Euclid: 23 September 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47A63: Operator inequalities
Secondary: 46L52: Noncommutative function spaces

Araki–Lieb–Thirring inequality von Neumann algebra $\tau$-measurable operator


Han, Yazhou. On the Araki–Lieb–Thirring inequality in the semifinite von Neumann algebra. Ann. Funct. Anal. 7 (2016), no. 4, 622--635. doi:10.1215/20088752-3660864.

Export citation


  • [1] M. Alakhrass, Inequalities related to Heinz mean, Linear Multilinear Algebra 64 (2015), no. 8, 1562–1569.
  • [2] H. Araki, On an inequality of Lieb and Thirring, Lett. Math. Phys. 19 (1990), no. 2, 167–170.
  • [3] T. N. Bekjan and D. Dauitbek, Submajorization inequalities of $\tau$-measurable operators for concave and convex functions, Positivity 19 (2015), no. 2, 341–345.
  • [4] T. N. Bekjan and G. Massen, Submajorization of the Araki–Lieb–Thirring inequality, Kyushu J. Math. 69 (2015), no. 2, 387–392.
  • [5] A. M. Bikchentaev, On normal $\tau$-measurable operators affliliated with semifinite von Neumann algebras, Math. Notes 96 (2014), no. 3–4, 332–341.
  • [6] J. C. Bourin and E. Y. Lee, Matrix inequalities from a two variables functional, preprint, arXiv:1511.06977.
  • [7] P. G. Dodds and T. K. Dodds, Some aspects of the theory of symmetric operator spaces, Quaest. Math. 18 (1995), no. 1–3, 47–89.
  • [8] P. G. Dodds, T. K. Dodds, P. N. Dowling, C. J. Lennard, and F. A. Sukochev, A uniform Kadec-Klee property for symmetric operator spaces, Math. Proc. Canib. Phil. Soc. 118 (1995), no. 3, 487–502.
  • [9] P. G. Dodds, T. K. Dodds, and F. A. Sukochev, On p-convexity and q-concavity in noncommutative symmetric spaces, Integral Equations Operator Theory 78 (2014), no. 1, 91–114.
  • [10] P. G. Dodds and F. A. Sukochev, Submajorisation inequalities for convex and concave functions of sums of measurable operators, Positivity 13 (2009), no. 1, 107–124.
  • [11] T. Fack, Sur la notion de valeur caracteristique, J. Operator Theory 7 (1982), no. 2, 307–333.
  • [12] T. Fack and H. Kosaki, Generalized s-numbers of $\tau$-measurable operators, Pac. J. Math. 123 (1986), no. 2, 269–300.
  • [13] F. Hansen, An operator inequality, Math. Ann. 246 (1980), no. 3, 249–250.
  • [14] F. Hiai, A generalization of Araki’s log-majorization, Linear Algebra Appl. 501 (2016), 1–16.
  • [15] H. Kosaki, Applications of uniform convexity of noncommutative $L^{p}$-spaces, Trans. Amer. Math. Soc. 283 (1984), no. 1, 265–282.
  • [16] H. Kosaki, An inequality of Araki–Lieb–Thirring (von Neumann algebra case), Proc. Amer. Math. Soc. 114 (1992), no. 2, 477–481.
  • [17] E. Lieb and W. Thirring, “Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities” in Studies in Mathematical Physics: Essays in Honor of Valentine Bargmann, Princeton Univ. Press, Princeton, 1976, 269–303.
  • [18] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces, II: Function Spaces, Ergeb. Math. Grenzgeb. 97, Springer, Berlin, 1979.
  • [19] Y. Nakamura, An inequality for generalized s-numbers, Integral Equations Operator Theory 10 (1987), no. 1, 140–145.
  • [20] G. Pisier and Q. Xu, “Noncommutative $L^{p}$-spaces” in Handbook of the Geometry of Banach Spaces, Vol. 2, North-Holland, Amsterdam, 2003, 1459–1517.
  • [21] M. Terp, $L^{p}$ spaces associated with von Neumann algebras, lecture notes, Univ. of Copenhagen, Copenhagen, 1981.
  • [22] Q. Xu, Analytic functions with values in lattices and symmetric spaces of measurable operators, Math. Proc. Camb. Phil. Soc. 109 (1991), no. 3, 541–563.