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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
73/2018

Curious properties of hypergraph C*-algebras

Tobias Fritz

Abstract

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.

Received:
Aug 27, 2018
Published:
Aug 29, 2018
MSC Codes:
46L99, 03D80, 81P13, 03F40
Keywords:
hypergraph C*-algebra, undecidability, Connes embedding problem, nonlocal games, first incompleteness theorem

Related publications

inJournal
2020 Repository Open Access
Tobias Fritz

Curious properties of free hypergraph C*-algebras

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