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Journal of Operator Theory

Volume 83, Issue 2, Spring 2020  pp. 423-445.

Curious properties of free hypergraph $C^*$-algebras

Authors:  Tobias Fritz
Author institution:Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany

Summary: 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 $C^*(H) \neq 0$ 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).

Keywords: free hypergraph $C^*$-algebra, undecidability, Connes embedding problem, nonlocal games, first incompleteness theorem

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