Commutative Counterexamples to Arveson's Hyperrigidity Conjecture

Abstract

We construct counterexamples to Arveson's hyperrigidity conjecture for operator systems in commutative $C^*$-algebras. In each example, every irreducible representation of the $C^*$-envelope is a boundary representation, but hyperrigidity fails. The first example is the operator system of continuous affine functions on an infinite-dimensional weighted $\ell^p$ ball. In the second infinite-dimensional example, we start with commuting operators and construct a nontrivial dilation. The final example comes from a compact truncated epigraph of a strictly convex function and yields a finite-dimensional operator system.