A new obstruction to Arveson's hyperrigidity conjecture

Abstract Let A be a unital $\mathrm {C}^*$ -algebra containing a closed two-sided ideal J and an operator system X . We enlarge X to an operator system $\mathcal{S} (X,J)$ in ${\mathbb {M}}_2(A)$ , and show that in order for $\mathcal{S} (X,J)$ to be hyperrigid, each $*$ -representation of $\mathrm {C}^*(X)$ annihilating $\mathrm {C}^*(X)\cap J$ must admit a unique contractive completely positive extension from X to the larger $\mathrm {C}^*$ -algebra $\mathrm {C}^*(X)+J$ . We leverage this implicit additional rigidity constraint to construct counterexamples to Arveson’s hyperrigidity conjecture. A key condition in our construction is the mutual orthogonality of the atomic projection of $\mathrm {C}^*(X)$ and the support projection of J , which we interpret as a new obstruction to the conjecture. Specializing to the case where J is the ideal of compact operators on a Hilbert space, we recover as a by-product of our general construction the recent counterexample of Bilich and Dor-On. On the other hand, we find that such a pathology cannot be implemented using our construction when A admits only finite-dimensional irreducible $*$ -representations, thereby illustrating that the obstruction only manifests itself in noncommutative settings.