Thomas Sinclair

math.OA — Operator Algebras

The regular pentagonal system and the Smith--Ward problem

Contributed by Shengwei Qiu, Thomas Sinclair

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 graph operator system and its dual are C$^*$-nuclear when the graph is $2$-chordal; otherwise they detect nuclearity and WEP, respectively.