  We consider separately radial (with corresponding group $\mathbb{T}^n$) and
radial (with corresponding group $\mathrm{U}(n))$ symbols on the projective
space $\mathbb{P}^n(\mathbb{C})$, as well as the associated Toeplitz operators
on the weighted Bergman spaces. It is known that the $C^*$-algebras generated
by each family of such Toeplitz operators are commutative. We present a new
representation theoretic proof of such commutativity. Our method is easier and
more enlightening as it shows that the commutativity of the $C^*$-algebras is a
consequence of the existence of multiplicity-free representations. Furthermore,
our method shows how to extend the current formulas for the spectra of the
corresponding Toeplitz operators to any closed group lying between
$\mathbb{T}^n$ and $\mathrm{U}(n)$.
