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Jensen’s inequality for spectral order and submajorization. (English) Zbl 1121.46042

Summary: Let \({\mathcal A}\) be a \(C^*\)-algebra and \(\varphi:{\mathcal A}\to L(H)\) be a positive unital map. Then, for a convex function \(f:I\to\mathbb{R}\) defined on some open interval and a selfadjoint element \(a\in{\mathcal A}\) whose spectrum lies in \(i\), we obtain a Jensen-type inequality \(f(\varphi(a))\leq\varphi(f(a))\), where \(\leq\) denotes an operator preorder (usual order, spectral preorder, majorization) and depends on the class of convex functions considered, i.e., monotone convex or arbitrary convex functions. Some extensions of Jensen-type inequalities to the multivariable case are considered.

MSC:

46L05 General theory of \(C^*\)-algebras
47A63 Linear operator inequalities
26D99 Inequalities in real analysis
39B72 Systems of functional equations and inequalities
39B52 Functional equations for functions with more general domains and/or ranges
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References:

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