Arveson’s hyperrigidity conjecture for function systems in commutative C*-algebras

Determine whether Arveson’s hyperrigidity conjecture holds for every function system S contained in a commutative C*-algebra: specifically, establish whether the condition that all irreducible representations of the C*-envelope C^*_{env}(S) are boundary representations implies that S is hyperrigid.

Background

Arveson’s hyperrigidity conjecture posits that if every irreducible representation of the C*-envelope of an operator system is a boundary representation, then the operator system is hyperrigid. While a recent counterexample shows the conjecture fails in general, important subclasses may still satisfy it.

Function systems (operator systems contained in commutative C*-algebras) are a natural setting for testing the conjecture, connecting noncommutative Choquet theory with classical function theory. The paper highlights that, despite general counterexamples, the status of the conjecture for function systems remains unresolved.