Title: Completely bounded bimodule maps and spectral synthesis,
Abstract: We initiate the study of the completely bounded multipliers of the Haagerup
tensor product $A(G)\otimes_{\rm h} A(G)$ of two copies of the Fourier algebra
$A(G)$ of a locally compact group $G$. If $E$ is a closed subset of $G$ we let
$E^{\sharp} = \{(s,t) : st\in E\}$ and show that if $E^{\sharp}$ is a set of
spectral synthesis for $A(G)\otimes_{\rm h} A(G)$ then $E$ is a set of local
spectral synthesis for $A(G)$. Conversely, we prove that if $E$ is a set of
spectral synthesis for $A(G)$ and $G$ is a Moore group then $E^{\sharp}$ is a
set of spectral synthesis for $A(G)\otimes_{\rm h} A(G)$. Using the natural
identification of the space of all completely bounded weak* continuous
$VN(G)'$-bimodule maps with the dual of $A(G)\otimes_{\rm h} A(G)$, we show
that, in the case $G$ is weakly amenable, such a map leaves the multiplication
algebra of $L^{\infty}(G)$ invariant if and only if its support is contained in
the antidiagonal of $G$.