Completely bounded bimodule maps and spectral synthesis

We initiate the study of the completely bounded multipliers of the Haagerup tensor product A(G)⊗hA(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 ♯ = {(s, t) : st ∈ E} and show that if E ♯ is a set of spectral synthesis for A(G) ⊗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 ♯ is a set of spectral synthesis for A(G) ⊗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) ⊗h A(G), we show that, in the case G is weakly amenable, such a map leaves the multiplication algebra of L ∞(G) invariant if and only if its support is contained in the antidiagonal of G.