  This paper is devoted to the study of pointwise convergence of Fourier series
for group von Neumann algebras and quantum groups. It is well-known that a
number of approximation properties of groups can be interpreted as summation
methods and mean convergence of the associated noncommutative Fourier series.
Based on this framework, this paper studies the refined counterpart of
pointwise convergence of these Fourier series. As a key ingredient, we develop
a noncommutative bootstrap method and establish a general criterion of maximal
inequalities for approximate identities of noncommutative Fourier multipliers.
Based on this criterion, we prove that for any countable discrete amenable
group, there exists a sequence of finitely supported positive definite
functions tending to $1$ pointwise, so that the associated Fourier multipliers
on noncommutative $L_p$-spaces satisfy the pointwise convergence for all $p>1$.
In a similar fashion, we also obtain results for a large subclass of groups (as
well as quantum groups) with the Haagerup property and the weak amenability. We
also consider the analogues of Fej\'{e}r and Bochner-Riesz means in the
noncommutative setting. Our approach heavily relies on the noncommutative
ergodic theory in conjunction with abstract constructions of Markov semigroups,
inspired by quantum probability and geometric group theory. Finally, we also
obtain as a byproduct the dimension free bounds of the noncommutative
Hardy-Littlewood maximal inequalities associated with convex bodies.
