Gruss inequality for some types of positive linear maps

Abstract: Assuming a unitarily invariant norm $|||\cdot|||$ is given on a two-sided ideal of bounded linear operators acting on a separable Hilbert space, it induces some unitarily invariant norms $|||\cdot|||$ on matrix algebras $\mathcal{M}n$ for all finite values of $n$ via $|||A|||=|||A\oplus 0|||$. We show that if $\mathscr{A}$ is a $C^*$-algebra of finite dimension $k$ and $\Phi: \mathscr{A} \to \mathcal{M}_n$ is a unital completely positive map, then \begin{equation*} |||\Phi(AB)-\Phi(A)\Phi(B)||| \leq \frac{1}{4} |||I{n}|||\,|||I_{kn}||| d_A d_B \end{equation*} for any $A,B \in \mathscr{A}$, where $d_X$ denotes the diameter of the unitary orbit ${UXU^*: U \mbox{ is unitary}}$ of $X$ and $I_{m}$ stands for the identity of $\mathcal{M}_{m}$. Further we get an analogous inequality for certain $n$-positive maps in the setting of full matrix algebras by using some matrix tricks. We also give a Gr\"uss operator inequality in the setting of $C^*$-algebras of arbitrary dimension and apply it to some inequalities involving continuous fields of operators.