Matrix Duality and a Bipolar Theorem for Completely Positive Maps
Mohsen Kian
TL;DR
The paper develops a matrix bipolar framework for completely positive maps between operator systems by pairing CP maps with matrix tests $(k,f,s)$ and introducing the saturated polar. It proves the main result $( ext{K}^{ riangle})^{ heta_{C^*}} = ar{ ext{cconv}( ext{K})}^{ au}$, linking the $C^*$-convex hull of a map family to a $ au$-closed polar; the topology $ au$ is generated by functionals $p_{k,f,s}(oldsymbol{ lux}) = |f(oldsymbol{ lux}_k(s))|$. This unifies matrix convex, $C^*$-convex, and tracial dualities, recovering Effros–Winkler in the matrix setting and Helton–Klep–McCullough in the tracial setting, with the finite-dimensional case aligning via the Jamiołkowski correspondence. The results provide a complete dual characterization of $C^*$-convex subsets of $ ext{CP}(\mathscr{S},oldsymbol{\,oldsymbol{ ext{T}}})$ and extend classical bipolar theory to a morphism-level, noncommutative context.
Abstract
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 $Φ:\mathscr{S}\to \mathscr{T}$ 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 $τ$--closed $C^*$--convex hull of $\mathcal K$, where $τ$ 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 Jamiołkowski isomorphism, to the matrix bipolar theorem for cones of positive semidefinite matrices.