Characterization of state-extension equality under positive epimorphisms

Characterize the conditions under which the transpose of a unital positive surjective map between unital C*-algebras maps a specified weakly compact convex set of states onto the corresponding specified state set, that is, determine when $\Psi^{\rm t}(\mathcal S_2)=\mathcal S_1$.

Background

The authors formulate a general state-extension problem for a unital positive surjective map Ψ between two unital C*-algebras and weakly compact convex subsets of their state spaces. Although the transpose is always injective and state-preserving, equality of its image with the prescribed state set is not automatic. This general issue motivates the later comparison of symmetric states on maximal, minimal, and other completions of twisted infinite chains.