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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.
47A63 Linear operator inequalities
46L10 General theory of von Neumann algebras
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