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MiS Preprint

Curious properties of hypergraph C*-algebras

Tobias Fritz


Given a finite hypergraph $H$, the associated hypergraph C*-algebra $C^*(H)$ is finitely presented by one projection for each vertex of $H$, such that each hyperedge forms a partition of unity. General hypergraph C*-algebras were first studied in the context of quantum contextuality, and there is no direct relation to graph C*-algebras. As special cases, the class of hypergraph C*-algebras comprises quantum permutation groups, maximal group C*-algebras of graph products of finite cyclic groups, and the C*-algebras associated to quantum graph homomorphism, isomorphism, and colouring.

Here, we conduct the first systematic study of aspects of hypergraph C*-algebras. We show that they coincide with the class of finite colimits of finite-dimensional commutative C*-algebras, and also with the class of C*-algebras associated to synchronous nonlocal games. We had previously shown that it is undecidable to determine whether $C^*(H)$ is nonzero for given $H$. We now show that it is also undecidable to determine whether a given $C^*(H)$ is residually finite-dimensional, and similarly whether it only has infinite-dimensional representations, and whether it has a tracial state. It follows that for each one of these properties, there is $H$ such that the question whether $C^*(H)$ has this property is independent of the ZFC axioms, assuming that these are consistent. We clarify some of the subtleties associated with such independence results in an appendix.

MSC Codes:
46L99, 03D80, 81P13, 03F40
hypergraph C*-algebra, undecidability, Connes embedding problem, nonlocal games, first incompleteness theorem

Related publications

2020 Repository Open Access
Tobias Fritz

Curious properties of free hypergraph C*-algebras

In: Journal of operator theory, 83 (2020) 2, pp. 423-445