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Sharp uniform convexity and smoothness inequalities for trace norms. (English) Zbl 0803.47037
Summary: We prove several sharp inequalities specifying the uniform convexity and uniform smoothness properties of the Schatten trace ideals \(C_ p\), which are the analogs of the Lebesgue spaces \(L_ p\) in non-commutative integration. The inequalities are all precise analogs of results which had been known in \(L_ p\), but were only known in \(C_ p\) for special values of \(p\). In the course of our treatment of uniform convexity and smoothness inequalities for \(C_ p\) we obtain new and simple proofs of the known inequalities for \(L_ p\).

47L20 Operator ideals
46B20 Geometry and structure of normed linear spaces
46L51 Noncommutative measure and integration
46L53 Noncommutative probability and statistics
46L54 Free probability and free operator algebras
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[1] [ArYa] Araki, H., Yamagami, S.: An inequality for the Hilbert-Schmidt norm. Commun. Math. Phys.81, 89-96 (1981) · Zbl 0468.47013 · doi:10.1007/BF01941801
[2] [BP] Ball, K., Pisier, G.: Unpublished result; private communication.
[3] [Bo] Boas, R.P.: Some uniformly convex spaces. Bull. Am. Math. Soc.46, 304-311 (1940) · Zbl 0024.41304 · doi:10.1090/S0002-9904-1940-07207-6
[4] [C] Clarkson, J.A.: Uniformly convex spaces. Trans. Am. Math. Soc.40, 396-414 (1936) · JFM 62.0460.04 · doi:10.1090/S0002-9947-1936-1501880-4
[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) · Zbl 0796.46054 · doi:10.1007/BF02100048
[6] [D] Day, M.: Uniform convexity in factor and conjugate spaces. Ann. Math.45, 375-385 (1944) · Zbl 0063.01058 · doi:10.2307/1969275
[7] [Di] Dixmier, J.: Formes linéaires sur un anneau d’opérateurs. Bull. Soc. Math. Fr.81, 222-245 (1953)
[8] [F] Figiel, T.: On the moduli of convexity and smoothness. Studia Math.56, 121-155 (1976) · Zbl 0344.46052
[9] [FJ] Figiel, T., Johnson, S.B.: A uniformly convex Banach space which contains noC p . Compos. Math.29, 179-190 (1974) · Zbl 0301.46013
[10] [Gr] Gross, L.: Logarithmic Sobolev inequalities. Am. J. Math.97, 1061-1083 (1975) · Zbl 0318.46049 · doi:10.2307/2373688
[11] [H] Hanner, O.: On the uniform convexity ofL p andl p . Ark. Math.3, 239-244 (1956) · Zbl 0071.32801 · doi:10.1007/BF02589410
[12] [Kö] Köthe, G.: Topologische lineare Räume, Die Grundlehren der mathematischen Wissen schaften in Einzeldarstellungen, Bd. 107, Springer Berlin Heidelberg New York: 1960
[13] [L] Lindenstrauss, J.: On the modulus of smoothness and divergent series in Banach spaces. Mich. Math. J.10, 241-252 (1963) · Zbl 0115.10001 · doi:10.1307/mmj/1028998906
[14] [P] Pisier, G.: The volume of convex bodies and Banach space geometry. Cambridge: Cambridge University Press, 1989 · Zbl 0698.46008
[15] [Ru] Ruskai, M.B.: Inequalities for traces on Von Neumann algebras. Commun. Math. Phys.26, 280-289 (1972) · Zbl 0257.46101 · doi:10.1007/BF01645523
[16] [Se] Segal, I.E.: A non-commutative extension of abstract integration. Ann. Math.57, 401-457 (1953) · Zbl 0051.34201 · doi:10.2307/1969729
[17] [Si] Simon, B.: Trace ideals and their applications. (See p. 22) Cambridge: Cambridge University Press, 1979
[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) · Zbl 0282.46016
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