  A notion of L^2-homology for compact quantum groups is introduced,
generalizing the classical notion for countable, discrete groups. If the
compact quantum group in question has tracial Haar state, it is possible to
define its L^2-Betti numbers and Novikov-Shubin invariants/capacities. It is
proved that these L^2-Betti numbers vanish for the Gelfand dual of a compact
Lie group and that the zeroth Novikov-Shubin invariant equals the dimension of
the underlying Lie group. Finally, we relate our approach to the approach of A.
Connes and D. Shlyakhtenko by proving that the L^2-Betti numbers of a compact
quantum group, with tracial Haar state, are equal to the Connes-Shlyakhtenko
L^2-Betti numbers of its Hopf *-algebra of matrix coefficients.
