Kadison duality for biaffinely convex sets

Abstract

Classical Kadison duality establishes a categorical equivalence between compact convex sets and Archimedean order unit spaces of affine functions. This paper develops a Kadison-type duality for sets with coordinate-wise convex structure in product domains $\mathbb{R}^n \times \mathbb{R}^m$. While slice-wise convex (biconvex) sets appear to be the natural geometric domain, separate convexity along coordinates is insufficient to ensure separation by biaffine polynomials. In classical convexity, every closed convex set is determined by affine functionals because any exterior point can be separated by a hyperplane. In contrast, biaffine polynomials can only probe a biconvex set $K$ up to its biaffine hull biconv$(K)$, the envelope formed by the intersection of all regions cut out by valid biaffine inequalities on $K$. To resolve this obstruction, the geometric setting is restricted to biaffinely convex sets, those that coincide with their biaffine hull ($K =$ biconv$(K)$) and therefore admit strict biaffine separation. On the algebraic side, we introduce the dual objects, which we call biaffine order unit spaces. These are finite-dimensional Archimedean order unit spaces defined on tensor products $E_1 \otimes E_2$ whose order units factor and whose positive cones are governed entirely by unentangled, rank-1 tensor evaluations. We prove a duality between the category of compact non-degenerate biaffinely convex sets with coordinate-wise affine maps and the category of abstract biaffine order unit spaces with tensor-factorized unital positive linear maps. In particular, every such biaffinely convex set is isomorphic to the coordinate state space of its biaffine polynomials, and conversely, every biaffine order unit space is order-isomorphic to the space of biaffine polynomials on its coordinate state space.