We establish a matrix bipolar theorem for families of completely positive maps between operator systems. Given operator systems $\mathscr{ S}$ and $\mathscr{ T}$, we introduce a canonical pairing between CP maps $\Phi:\mathscr{ S}\to \mathscr{ T}$ and \emph{ matrix tests} $(k,f,s)$, where $k\in\mathbb{ N}$, $f$ is a state on $M_k(\mathscr{ T})$, and $s\in M_k(\mathscr{ S})$. This pairing induces a duality between subsets of $\CP(\mathscr{ S},\mathscr{ T})$ and collections of matrix tests through a \emph{ saturated polar} construction that preserves $C^*$--convexity. Our main theorem identifies the double polar $(\mathcal K^{ \circ_{ C^*}})^{ \circ_{ C^*}}$ with the $\tau$--closed $C^*$--convex hull of $\mathcal K$, where $\tau$ is the locally convex topology generated by the test functionals. The result extends the classical matrix bipolar theorem of Effros--Winkler for matrix-convex sets (the case $\mathscr{ T}=\mathbb{ C}$) and the tracial bipolar theorem of Helton--Klep--McCullough (for tracial $\mathscr{ T}$), while providing the first complete dual characterization of $C^*$--convex subsets of $\CP(\mathscr{ S},\mathscr{ T})$. In finite dimensions, the theorem corresponds, via the Jamio{ \l}kowski isomorphism, to the matrix bipolar theorem for cones of positive semidefinite matrices.
Categories: math.OA math.FA