Hyperrigidity I: singly generated commutative $C^*$-algebras

Although Arveson's hyperrigidity conjecture was recently resolved negatively by B. Bilich and A. Dor-On, the problem remains open for commutative $C^*$-algebras. Relatively few examples of hyperrigid sets are known in the commutative case. The main goal of this paper is to determine which sets of monomials in $t$ and $t^*$, where $t$ is a generator of a commutative unital $C^*$-algebra, are hyperrigid. We show that this class of hyperrigid sets has significant connections to other areas of functional analysis and mathematical physics. Moreover, we develop a topological approach based on weak and strong limits of normal (or subnormal) operators to characterize hyperrigidity tracing back to ideas of C. Kleski and L. G. Brown. Employing Choquet boundary techniques, we present examples that discuss the optimality of our results.