Gruss inequality for some types of positive linear maps

Assuming a unitarily invariant norm $\mathrm{\mid }\mathrm{\mid }\mathrm{\mid }\cdot \mathrm{\mid }\mathrm{\mid }\mathrm{\mid }$ is given on a two-sided ideal of bounded linear operators acting on a separable Hilbert space, it induces some unitarily invariant norms $\mathrm{\mid }\mathrm{\mid }\mathrm{\mid }\cdot \mathrm{\mid }\mathrm{\mid }\mathrm{\mid }$ on matrix algebras ${\mathcal{M}}_n$ for all finite values of $n$ via $\mathrm{\mid }\mathrm{\mid }\mathrm{\mid }A\mathrm{\mid }\mathrm{\mid }\mathrm{\mid }=\mathrm{\mid }\mathrm{\mid }\mathrm{\mid }A\oplus 0\mathrm{\mid }\mathrm{\mid }\mathrm{\mid }$. We show that if $\mathcal{A}$ is a $C^{\ast }$-algebra of finite dimension $k$ and $\mathrm{\Phi }:\mathcal{A}\to {\mathcal{M}}_n$ 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$d_X$denotesthediameteroftheunitaryorbit${UXU^*: U \mbox{ is unitary}}$of$X$and$I_{m}$standsfortheidentityof$$\mathcal{M}$_{m}$\mathrm{.}Furtherwegetananalogousinequalityforcertain$n$-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 $C^{\ast }$-algebras of arbitrary dimension and apply it to some inequalities involving continuous fields of operators.