  We continue the investigation of notions of approximate amenability that were
introduced in work of the second and third authors. It is shown that every
boundedly approximately contractible Banach algebra has a bounded approximate
identity.
  Among our other results, it is shown that the Fourier algebra of the free
group on two generators is not approximately amenable. Further examples are
obtained of ${\ell}^1$-semigroup algebras which are approximately amenable but
not amenable; using these, we show that bounded approximate amenability need
not imply sequential approximate amenability. Results are also given for Segal
subalgebras of $L^1(G)$, where $G$ is a locally compact group, and the algebras
$PF_p(\Gamma)$ of $p$-pseudofunctions on a discrete group $\Gamma$ (of which
the reduced $C^*$-algebra is a special case).
