Geodesic
To the theory of operator monotone and operator convex functions
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 3 (2010), pp. 9-14.

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We prove that a real function is operator monotone (operator convex) if the corresponding monotonicity (convexity) inequalities are valid for some normal state on the algebra of all bounded operators in an infinite-dimensional Hilbert space. We describe the class of convex operator functions with respect to a given von Neumann algebra in dependence of types of direct summands in this algebra. We prove that if a function from $\mathbb R^+$ into $\mathbb R^+$ is monotone with respect to a von Neumann algebra, then it is also operator monotone in the sense of the natural order on the set of positive self-adjoint operators affiliated with this algebra.
Keywords: operator monotone function, operator convex function, von Neumann algebra, $C^*$-algebra. 
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Dinh Trung Hoa; O. E. Tikhonov. To the theory of operator monotone and operator convex functions. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 3 (2010), pp. 9-14. http://geodesic.mathdoc.fr/item/IVM_2010_3_a1/

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