Completely bounded bimodule maps and spectral synthesis

Abstract

We initiate the study of the completely bounded multipliers of the Haagerup tensor product $A(G){\otimes }_{\mathrm{h}}A(G)$ 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 $E^{♯}=\{(s,t):st\in E\}$ and show that if $E^{♯}$ is a set of spectral synthesis for $A(G){\otimes }_{\mathrm{h}}A(G)$ 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 $A(G){\otimes }_{\mathrm{h}}A(G)$. Using the natural identification of the space of all completely bounded weak* continuous $VN(G{)}^{\prime }$-bimodule maps with the dual of $A(G){\otimes }_{\mathrm{h}}A(G)$, 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 $G$.

44 pages

References in corpus (1)

• Amenability and weak amenability of the Fourier algebra