Completely bounded bimodule maps and spectral synthesis

Journal article, 2017

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