Curious properties of free hypergraph C*algebras
Abstract
A finite hypergraph $H$ consists of a finite set of vertices $V(H)$ and a collection of subsets $E(H) \subseteq 2^{V(H)}$ which we consider as partition of unity relations between projection operators. These partition of unity relations freely generate a universal C*algebra, which we call the "free hypergraph C*algebra" $C^*(H)$. General free hypergraph C*algebras were first studied in the context of quantum contextuality. As special cases, the class of free 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 free hypergraph C*algebras. We show that they coincide with the class of finite colimits of finitedimensional 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 finitedimensional, and similarly whether it only has infinitedimensional 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.
 Publication:

arXiv eprints
 Pub Date:
 August 2018
 DOI:
 10.48550/arXiv.1808.09220
 arXiv:
 arXiv:1808.09220
 Bibcode:
 2018arXiv180809220F
 Keywords:

 Mathematics  Operator Algebras;
 Mathematics  Logic;
 Quantum Physics;
 46L99;
 03D80 (Primary);
 81P13;
 03F40 (Secondary)
 EPrint:
 19 pages. v2: minor clarifications. v3: terminology 'free hypergraph C*algebra', added Remark 2.21