A new obstruction to Arveson's hyperrigidity conjecture

Abstract

Let $A$ be a unital $C^*$-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 $C^*(X)$ annihilating $C^*(X)\cap J$ must admit a unique contractive completely positive extension from $X$ to the larger $C^*$-algebra $C^*(X)+J$. 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 $C^*(X)$ 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.