On the equality in Jensen's inequality for operator convex functions
Let \(A,B\) be unital \(C^*\)-algebras, and let \(\Phi:A\to B\) be a positive unit preserving linear map. \textit{M. D. Choi} [Ill. J. Math. 18, 565-574 (1974; Zbl 0293.46043)] has proved that \(f(\Phi(a))\leq \Phi(f(a))\) for all self-adjoint \(a\in A\) with \(Sp(a)\subset (\alpha,\beta)\) and \(f:(\alpha,\beta)\to {\mathbb{R}}\) an operator convex function. This paper proves that equality holds if and only if \(\Phi\) restricted to the subalgebra generated by a is multiplicative.
scientific article; zbMATH DE number 4040587
scientific article