Curious properties of free hypergraph C*-algebras

TL;DR

This paper systematically studies free hypergraph C*-algebras, revealing their structural properties, connections to nonlocal games, and demonstrating the undecidability and independence of key algebraic properties within set theory.

Contribution

It provides the first comprehensive analysis of free hypergraph C*-algebras, establishing their equivalence with finite colimits of finite-dimensional commutative C*-algebras and linking them to synchronous nonlocal games.

Findings

Coincide with finite colimits of finite-dimensional commutative C*-algebras

Undecidable to determine residual finite-dimensionality

Certain properties are independent of ZFC axioms

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 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.