## Spectral synthesis and operator synthesis

### Volume 177 / 2006

#### Abstract

Relations between spectral synthesis in the Fourier algebra $A(G)$ of a compact group $G$ and the concept of operator synthesis due to Arveson have been studied in the literature. For an $A(G)$-submodule $X$ of $\mathop {\rm VN}\nolimits (G)$, $X$-synthesis in $A(G)$ has been introduced by E. Kaniuth and A. Lau and studied recently by the present authors. To any such $X$ we associate a $V^{\infty }(G)$-submodule $ \widehat {X}$ of ${\mathcal B}(L^{2}(G))$ (where $V^{\infty }(G)$ is the weak-$*$ Haagerup tensor product $L^{\infty }(G)\otimes _{w^{*}h} L^{\infty }(G)$ ), define the concept of $ \widehat {X}$-operator synthesis and prove that a closed set $E$ in $G$ is of $X$-synthesis if and only if $E^{*}:=\{ (x,y)\in G\times G: xy^{-1}\in E\} $ is of $\widehat {X}$-operator synthesis.