‎In this paper‎, ‎we aim to replace in the definitions of covariance and correlation the usual trace {\\\\\\\\\\\\\\\\rm Tr} by a tracial positive map between unital $C^*$-algebras and to ‎replace ‎the ‎functions‎ $x^{\\\\\\\\\\\\\\\\alpha}$ and $x^{1-\\\\\\\\\\\\\\\\alpha}$ by functions $f$ and $g$ satisfying some mild conditions‎. ‎These allow us to define the generalized covariance‎, ‎the generalized variance‎, ‎the generalized correlation and the generalized Wigner--Yanase--Dyson skew information related to the tracial positive maps and functions $f$ and $g$‎. ‎We present a generalization of Heisenberg\\\\\\\\\\\\\\\'s uncertainty relation in the noncommutative framework‎. ‎We extend some inequalities and properties for the generalized correlation and the generalized Wigner--Yanase--Dyson skew information‎. ‎Furthermore‎, ‎we extend some inequalities for the generalized skew information such as uncertainty relation and the relation between the generalized variance and the generalized skew information‎.

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