  In order to investigate the relationship between weak amenability and the
Haagerup property for groups, we introduce the weak Haagerup property, and we
prove that having this approximation property is equivalent to the existence of
a semigroup of Herz-Schur multipliers generated by a proper function. It is
then shown that a (not necessarily proper) generator of a semigroup of
Herz-Schur multipliers splits into a positive definite kernel and a
conditionally negative definite kernel. We also show that the generator has a
particularly pleasant form if and only if the group is amenable.
  In the second half of the paper we study semigroups of radial Herz-Schur
multipliers on free groups. We prove that a generator of such a semigroup is
linearly bounded by the word length function.
