  We study analogues of the radial subalgebras in free group factors (called
the algebras of class functions) in the setting of compact quantum groups. For
the free orthogonal quantum groups we show that they are not MASAs, as soon as
we are in a non-Kac situation. The most important notion to our present work is
that of a (quasi-)split inclusion. We prove that the inclusion of the algebra
of class functions is quasi-split for some unitary quantum groups; in this case
the subalgebra is non-abelian and we also obtain a result concerning its
relative commutant. In the positive direction, we construct certain bicrossed
products from the quantum group $SU_q(2)$ for which the algebra of class
functions is a MASA.
