Curious properties of hypergraph C*-algebras
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Submission date: 27. Aug. 2018
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.