Preprint 73/2018

Curious properties of hypergraph C*-algebras

Tobias Fritz

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Submission date: 27. Aug. 2018
Pages: 20
MSC-Numbers: 46L99, 03D80, 81P13, 03F40
Keywords and phrases: hypergraph C*-algebra, undecidability, Connes embedding problem, nonlocal games, first incompleteness theorem
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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.

24.11.2021, 02:20