  In this paper, we study hyperrigidity for singly generated unital commutative
$C^*$-algebras. We provide four main approaches to hyperrigidity. The first is
via operator-moment characterizations of spectral measures, the subject
intensively exploited in quantum theory. The second approach is based on
dilation theory and is written in terms of the Stone-von Neumann calculus for
normal operators. The third, inspired by a recent result of L. G. Brown,
concerns the weak and strong convergence of sequences of subnormal (or normal)
operators. Finally, the fourth approach deals with the multiplicativity of
unital completely positive maps on $C^*$-subalgebras generated by single normal
elements. This 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.
