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 $\mathcal{A}$ is a $C^{\ast }$-algebra of finite dimension $k$ and $\Phi :\mathcal{A}\to {\mathcal{M}}_n$ is a unital completely positive map, then \begin{equation*} |||Φ(AB)-Φ(A)Φ(B)||| \leq \frac{1}{4} |||I_{n}|||\,|||I_{kn}||| d_A d_B \end{equation*} for any $A,B\in \mathcal{A}$, 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 Grüss operator inequality in the setting of $C^{\ast }$-algebras of arbitrary dimension and apply it to some inequalities involving continuous fields of operators.

17 pages, to appear in J. Operator Theory (JOT)

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