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We initiate the study of the completely bounded multipliers of the Haageruptensor 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 ofspectral synthesis for $A(G)\otimes_{\rm h} A(G)$ then $E$ is a set of localspectral synthesis for $A(G)$. Conversely, we prove that if $E$ is a set ofspectral synthesis for $A(G)$ and $G$ is a Moore group then $E^{\sharp}$ is aset of spectral synthesis for $A(G)\otimes_{\rm h} A(G)$. Using the naturalidentification 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 showthat, in the case $G$ is weakly amenable, such a map leaves the multiplicationalgebra of $L^{\infty}(G)$ invariant if and only if its support is contained inthe antidiagonal of $G$.

Completely bounded bimodule maps and spectral synthesis