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

In this paper, we study hyperrigidity for singly generated commutative unital $C^*$-algebras. We look at hyperrigidity from four different points of view. The first is via operator-moment characterizations of spectral measures, the subject intensively exploited in quantum theory. The second is based on dilation theory and the Stone-von Neumann calculus for normal operators. The third, inspired by a recent result by L. G. Brown, concerns the weak and strong convergence of sequences of subnormal (or normal) operators. Finally, the fourth deals with the multiplicativity of unital completely positive maps on $C^*$-subalgebras generated by single normal elements. The latter is inspired by Petz's theorem and its generalizations established first by Arveson in the finite-dimensional case and then by L. G. Brown in general. The above machinery, together with sophisticated operator inequalities (Kadison's, Hansen's and Lieb-Ruskai's), makes it possible to identify certain subsets $G$ of the set $\{t^{*m}t^n\colon m, n = 0,1,2,\ldots\}$ that are hyperrigid in $C^*(t)$, the unital $C^*$-algebra generated by a single normal element $t$. Examples based on the Choquet boundary are included to confirm the optimality of our results.