Abstract
We study inequalities of the form
where τ is a trace on a von Neumann algebra or a C*-algebra, A and B are self-adjoint elements of the algebra in question, f and w are real-valued functions, and the “weight” function w is nonnegative.
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Dinh Trung Hoa and O. E. Tikhonov, “Weighted trace inequalities of monotonicity,” Lobachevskii J. Math. 26, 63–67 (2007).
J. Dixmier, Les C*—algèbres et leurs représentations (Gauthier-Villars, Paris, 1969; Nauka, Moscow, 1974).
M. Takesaki, Theory of Operator Algebras (Springer-Verlag, New York, NY, 1979), Vol. I.
L. T. Gardner, “An inequality characterizes the trace,” Canad. J. Math. 31(6), 1322–1328 (1979).
D. Petz and J. Zemánek, “Characterizations of the trace,” Linear Algebra Appl. 111, 43–52 (1988).
A. I. Stolyarov, O. E. Tikhonov, and A. N. Sherstnev, “Characterization of normal traces on von Neumann algebras by inequalities for the modulus,” Mat. Zametki 72(3), 448–454 (2002) [Math. Notes 72 (3–4), 411–416 (2002)].
O. E. Tikhonov, “Subadditivity inequalities in von Neumann algebras and characterization of tracial functionals,” Positivity 9(2), 259–264 (2005).
M. Takesaki, Theory of Operator Algebras, Vol. II, in Encyclopaedia Math. Sci. Vol. 125: Operator Algebras and Non-Commutative Geometry, 6 (Springer-Verlag, New York, 2003).
O. E. Tikhonov, “Convex functions and inequalities for a trace,” in Constructive Theory of Functions and Functional Analysis (Kazan. Gos. Univ., Kazan, 1987), Vol. 6, pp. 77–82 [in Russian].
L. G. Brown and H. Kosaki, “Jensen’s inequality in semi-finite von Neumann algebras,” J. Operator Theory 23(1), 3–19 (1990).
O. E. Tikhonov, “Trace inequalities for spaces in spectral duality,” Studia Math. 104(1), 99–110 (1993).
A. M. Bikchentaev and O. E. Tikhonov, “Characterization of the trace by monotonicity inequalities,” Linear Algebra Appl. 422(1), 274–278 (2007).
T. Sano and T. Yatsu, “Characterizations of tracial property via inequalities,” JIPAM. J. Inequal. Pure Appl. Math. 7(1) (2006), Article No. 36.
T. Ogasawara, “A theorem on operator algebras,” J. Sci. Hiroshima Univ. Ser. A 18, 307–309 (1955).
R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras, Vol. II: Advanced Theory, in Pure Appl. Math. (Academic Press, Orlando, Fl, 1986), Vol. 100.
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Original Russian Text © Dinh Trung Hoa, O. E. Tikhonov, 2010, published in Matematicheskie Zametki, 2010, Vol. 88, No. 2, pp. 193–200.
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Hoa, D.T., Tikhonov, O.E. Weighted monotonicity inequalities for traces on operator algebras. Math Notes 88, 177–182 (2010). https://doi.org/10.1134/S0001434610070175
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DOI: https://doi.org/10.1134/S0001434610070175