Abstract
Given a finite hypergraph H, the \textit{free hypergraph C∗-algebra} C∗(H) is freely generated by a projection for each vertex subject to partition of unity relations specified by the hyperedges. We prove that the class of free hypergraph C∗-algebras coincides with the finite colimits of finite-dimensional commutative C∗-algebras, and with the C∗-algebras of synchronous nonlocal games. It is known that determining whether C∗(H)≠0 for given H is undecidable. We prove that it is also undecidable to determine whether C∗(H) is RFD, whether C∗(H) has only infinite-dimensional representations, and whether it has a tracial state. For each of these properties, there is H such that whether C∗(H) has this property is independent of ZFC (if ZFC is consistent).
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