  We study an averaging procedure for completely bounded multipliers on a
locally compact quantum group with respect to a compact quantum subgroup. As a
consequence we show that central approximation properties of discrete quantum
groups are equivalent to the corresponding approximation properties of their
Drinfeld doubles. This is complemented by a discussion of the averaging of
Fourier algebra elements. We compare the biinvariant Fourier algebra of the
Drinfeld double of a discrete quantum group with the central Fourier algebra.
In the unimodular case these are naturally identified, but we show by
exhibiting a family of counter-examples that they differ in general.
