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The Johnson--Kadison--Ringrose theorem states that the Aron--Berner extensionof a multilinear map from $C^*$-algebras to second dual spaces is unique andseparately weak$^*$-weak$^*$-continuous. We show that every positivemultilinear map between $C^*$-algebras is jointly weak$^*$-weak$^*$-continuous.Utilizing this fact, we obtain joint weak$^*$-weak$^*$-continuity of theAron--Berner extension in the case of positive maps.This result allows us toprove that every tracial positive multilinear map and every tracial completelypositive (not necessarily linear or multilinear) map $\Phi$ between$C^*$-algebras, enjoy a decomposition $\Phi=\varphi_2 \circ \varphi_1$, inwhich $\varphi_1$ is a tracial positive linear map with the commutative rangeand $\varphi_2$ is a tracial completely positive map with commutative domain.As a first result, tracial positive multilinear maps are completely positive.As an application, we investigate the generalized variance and covariance inquantum mechanics via arbitrary positive maps. Among others, an uncertaintyrelation type inequality for commuting observables in a composite physicalsystem has been presented: if $ \Phi: \mathscr{A}\bigotimes \mathscr{B} \to\mathscr{C}$ is a unital positive linear map between unital $C^*$-algebras and$A\otimes I_\mathscr{B}$ and $ I_ \mathscr{A} \otimes B$ are compatibleobservables in the composite system, then \[ {\rm Var}_\Phi(A\otimesI_\mathscr{B}) {\rm Var}_\Phi(I_\mathscr{A}\otimes B) \geq \frac{1}{16} \frac{\left|\Phi([{\textbf{A}}, {\textbf{C}}])\Phi([{\textbf{B}},{\textbf{D}}])\right|^2}{(\|C-\alpha I_\mathscr{A}\|\|D-\beta I_\mathscr{B}\|)^2 }\] for all self-adjoint elements $ C\in\mathscr{A}$ and $D\in\mathscr{B}$ and every $\alpha, \beta \in \mathbb{C}$with nonzero $ \|C-\alpha I_\mathscr{A}\|$ and $\|D-\beta I_\mathscr{B}\|$.

Decomposition of tracial positive maps and their applications in quantum information