We investigate some inequalities related to Chebyshev and Gr\"uss inequalities regarding some unitarily invariant norms. 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 some operator inequalities in the framework of $C^*$-algebras of arbitrary dimension and apply it to some inequalities involving continuous fields of operators. operator inequalities in the framework of C-algebras of arbitrary dimension and apply it to some inequalities involving continuous fields of operators.

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