Curious properties of free hypergraph C*-algebras [article]

Tobias Fritz
2019 arXiv   pre-print
A finite hypergraph H consists of a finite set of vertices V(H) and a collection of subsets E(H) ⊆ 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
more » ... 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 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.
arXiv:1808.09220v3 fatcat:7qdctzsjs5ge5l6bxzykqp5d2q