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Let us consider the set of positive maps $\phi:\mathcal{B}(\mathcal{H})\rightarrow \mathcal{B}(\mathcal{H})$ ($\mathcal{H}$ is Hilbert space). Can we characterize all the maps which satisfies the following derivative like inequality. \begin{equation} \phi(a^*a)\leq\phi(a^*)a+a^*\phi(a)\quad \forall a\in \mathcal{B}(\mathcal{H}) \end{equation} This is weaker than derivative, in general. I want to know, how much is known about the maps of the above type (some dilation theorem, or representation theorem). Partial answers are also welcome, for example when $\mathcal{H}$ is finite dimensional. In particular in finite dimension can one give me some non trivial examples of such maps?

Moreover if we consider any map (not necessarily positive) satisfying the above inequality, is it known anything about them. Any reference suggestion comments are welcome. Advanced thanks for all helps.

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