  We investigate quantum group generalizations of various density results from
Fourier analysis on compact groups. In particular, we establish the density of
characters in the space of fixed points of the conjugation action on
$L^2(\mathbb{G})$, and use this result to show the weak* density and norm
density of characters in $ZL^{\infty}(\mathbb{G})$ and $ZC(\mathbb{G})$,
respectively. As a corollary, we partially answer an open question of
Woronowicz. At the level of $L^1(\mathbb{G})$, we show that the center
$\mathcal{Z}(L^1(\mathbb{G}))$ is precisely the closed linear span of the
quantum characters for a large class of compact quantum groups, including
arbitrary compact Kac algebras. In the latter setting, we show, in addition,
that $\mathcal{Z}(L^1(\mathbb{G}))$ is a completely complemented
$\mathcal{Z}(L^1(\mathbb{G}))$-submodule of $L^1(\mathbb{G})$.
