Geodesic
Weighted Monotonicity Inequalities for Traces on Operator Algebras
Matematičeskie zametki, Tome 88 (2010) no. 2, pp. 193-200.

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We study inequalities of the form $$ \tau(w(A)^{1/2}f(A)w(A)^{1/2})\le\tau(w(A)^{1/2}f(B)w(A)^{1/2}),\qquad A\le B, $$ where $\tau$ 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.
Keywords: von Neumann algebra, $C^*$-algebra, trace, monotonicity, Hermitian matrix, weight function, self-adjoint element, tracial functional, Borel function. 
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Dinh Trung Hoa; O. E. Tikhonov. Weighted Monotonicity Inequalities for Traces on Operator Algebras. Matematičeskie zametki, Tome 88 (2010) no. 2, pp. 193-200. http://geodesic.mathdoc.fr/item/MZM_2010_88_2_a3/

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