the theory of completely positive (CP) linear maps between Banach spaces of trace class operators on separable Hilbert spaces (the Schatten classes of order 1). Since $\T(\H)^*=\B(\H)$, the below results can be reformulated in terms of CP linear maps between algebras of all bounded operators on separable Hilbert spaces. Nevertheless, the use of the “predual picture” is more natural for representation of our results. The theory of CP linear maps between Banach spaces of trace class operators has important applications in mathematical physics, in particular, in the theory of open quantum systems, where CP trace-preserving linear maps called *quantum channels* play the role of dynamical maps (in the Schrodinger picture), while CP trace-non-increasing linear maps called *quantum operations* are essentially used in the theory of quantum measurements [@H-SCI; @Watrous; @Wilde].

For a CP linear map $\,\Phi:\T(\H_A)\rightarrow \T(\H_B)\,$ the Stinespring theorem (cf.[@St]) implies existence of a Hilbert space $\mathcal{H}_E$ and an operator $V_{\Phi}:\mathcal{H}_A\rightarrow\mathcal{H}_B\otimes\mathcal{H}_E$ such that $$\label{St-rep}
\Phi(\rho)=\mathrm{Tr}_{E}V_{\Phi}\rho V_{\Phi}^{*},\quad
\rho\in\mathfrak{T}(\mathcal{H}_A),$$ where ${\mathrm{Tr}}_E$ denotes the partial trace over $\H_E$. If $\Phi$ is trace-preserving (correspondingly, trace-non-increasing) then $V_{\Phi}$ is an isometry (correspondingly, contraction) [@H-SCI Ch.6].

The dual CP linear map $\,\Phi^*:\B(\H_B)\rightarrow \B(\H_A)\,$ has the corresponding representation $$\label{St-rep+}
\Phi^*(B)=V^*_{\Phi}[B\otimes I_E] V_{\Phi},\quad
B\in\mathfrak{B}(\mathcal{H}_B).$$

The norm of 