For a locally compact group
$G$
and
$1\,<\,p\,<\,\infty $
, let
${{A}_{p}}\left( G \right)$
be the Herz-Figà-Talamanca algebra and let
$P{{M}_{p}}\left( G \right)$
be its dual Banach space. For a Banach
${{A}_{p}}\left( G \right)$
-module
$X$
of
$P{{M}_{p}}\left( G \right)$
, we prove that the multiplier space
$\text{M}\left( {{A}_{p}}\left( G \right),{{X}^{*}} \right)$
is the dual Banach space of
${{Q}_{X}}$
, where
${{Q}_{X}}$
is the norm closure of the linear span
${{A}_{p}}\left( G \right)X\,\text{of}\,u\,f\,\text{for}\,u\,\in \,{{A}_{p}}\left( G \right)\,\text{and}\,f\,\in \,X$
in the dual of
$\text{M}\left( {{A}_{p}}\left( G \right),{{X}^{*}} \right)$
. If
$p\,=\,2$
and
$P{{F}_{p}}\left( G \right)\subseteq X$
, then
${{A}_{p}}\left( G \right)X$
is closed in
$X$
if and only if
$G$
is amenable. In particular, we prove that the multiplier algebra
$M{{A}_{p}}\left( G \right)\,\text{of}\,{{A}_{p}}\left( G \right)$
is the dual of
$Q$
, where
$Q$
is the completion of
${{L}^{1}}\left( G \right)$
in the
$||\cdot |{{|}_{M}}$
-norm.
$Q$
is characterized by the following:
$f\,\in \,Q$
if an only if there are
${{u}_{i}}\,\in \,{{A}_{p}}\left( G \right)$
and
${{f}_{i}}\in P{{F}_{p}}\left( G \right)\left( i=1,2,... \right)$
with
$\sum\nolimits_{i=1}^{\infty }{||}\,{{u}_{i}}\,|{{|}_{{{A}_{p}}\left( G \right)}}||fi|{{|}_{P{{F}_{p}}\left( G \right)}}\,<\,\infty $
such that
$f=\sum{_{i=1}^{\infty }\,{{u}_{i}}{{f}_{i}}}$
on
$M{{A}_{p}}\left( G \right)$
. It is also proved that if
${{A}_{p}}\left( G \right)$
is dense in
$M{{A}_{p}}\left( G \right)$
in the associated
${{w}^{*}}$
-topology, then the multiplier norm and
$||\cdot |{{|}_{{{A}_{p}}\left( G \right)}}$
-norm are equivalent on
${{A}_{p}}\left( G \right))$
if and only if
$G$
is amenable.