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On some versions of Jensen’s inequality on operator algebras. (English) Zbl 1053.47505
Extensions of the Jensen inequality to operator algebras are well-known. If $$f$$ is an operator convex function and $$\alpha$$ is a positive unital mapping between two operator algebras, then $$\alpha \bigl (f (a)\bigr) \geq f\bigl (\alpha (a)\bigr)$$ for a selfadjoint operator $$a$$. In the paper under review, a noncommutative polynomial $$W(x,y)$$ is considered and the operator inequality $$\alpha \bigl (W(a, a^*)\bigr) \geq W\bigl (\alpha (a), \alpha (a^*)\bigr)$$ is obtained under some conditions, for example, $$a \mapsto W(a,a^*)$$ is assumed to be convex. The technique of $$2 \times 2$$ operator matrices and Stinespring representation is familiar from earlier works on inequalities. A trace inequality of Jensen type is also treated.
##### MSC:
 47A63 Linear operator inequalities 46L10 General theory of von Neumann algebras
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##### References:
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