Geodesic
On operator monotone and operator convex functions
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 5 (2016), pp. 70-74

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We establish monotonicity and convexity criteria for a continuous function $f\colon\mathbb R^+\to\mathbb R$ with respect to any $C^*$-algebra. We obtain some estimates for noncompactness measure of $W^*$-algebra elements products differences. We also give a commutativity criterion for a positive $\tau$-measurable operator and a positive operator from a von Neumann algebra.
Keywords: Hilbert space, von Neumann algebra, $C^*$-algebra, $W^*$-algebra, operator monotone function, operator convex function, measure of noncompactness, trace, measurable operator, commutativity of operators. 
@article{IVM_2016_5_a4,
     author = {A. M. Bikchentaev},
     title = {On operator monotone and operator convex functions},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {70--74},
     publisher = {mathdoc},
     number = {5},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2016_5_a4/}
}
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A. M. Bikchentaev. On operator monotone and operator convex functions. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 5 (2016), pp. 70-74. http://geodesic.mathdoc.fr/item/IVM_2016_5_a4/