The regular pentagonal system and the Smith--Ward problem
The three-dimensional function system of the regular pentagon fails the lifting property, giving a counterexample to the operator-system formulation of the Smith--Ward problem with C$^*$-envelope $\mathbb C^5$. We construct unital $k$-positive maps from this fixed system into matrix algebras where distance from every ucp map is bounded below independently of $k$. We also prove that every regular polygon system with at least five vertices detects nuclearity, while its dual characterizes the weak expectation property. These tensor-product characterizations follow from a graph dichotomy: a…
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