Let \(G\) be a compact group and \(A(G)\) be its Fourier algebra as defined by P. Eymard [Bull. Soc. Math. Fr. 92, 181–236 (1964; Zbl 0169.46403)]. It was shown in [E. G. Effros and Z.-J. Ruan, J. Oper. Theory 50, No. 1, 131–156 (2003; Zbl 1036.46042)] that the Haagerup tensor product of the Fourier algebra \(A(G)\) with itself, \(A(G)\otimes^h A(G)\), is a Banach algebra. In the present paper, the authors prove that this Banach algebra is, in fact, semi-simple. They study the ranges of the maps of convolution \(u\otimes v\mapsto u\ast v\) and a ‘twisted’ convolution \(u\otimes v\mapsto u\ast \check{v}\) \((\check{u}(s)=u(s^{-1}))\) on the Haagerup tensor product of Fourier algebras \(A(G)\otimes^h A(G)\).
If \(\mathcal{H}\) is a fixed Hilbert space, the weak\(^\ast\) Haagerup tensor product \(\mathcal{CB}^{\sigma}(\mathcal{B}(\mathcal{H}))=\mathcal{B}(\mathcal{H})\otimes^{w^\ast h}\mathcal{B}(\mathcal{H})\) is the space \(\{\Phi\in\mathcal{B}(\mathcal{H}): \Phi(T)=\sum_{i\in I}V_i\otimes W_i(T)=\sum_{i\in I}V_iTW_i \}, \) equipped with the completely bounded norm \[ \|\Phi\|_{cb}=\min\bigg\{\bigg\|\displaystyle\sum_{i\in I}V_iV_{i}^\ast \bigg\|^{1/2}\bigg\|\displaystyle\sum_{i\in I}W_{i}^\ast W_i\bigg\|^{1/2}: \Phi=\displaystyle\sum_{i\in I}V_{i}\otimes W_{i} \bigg\} \] and operator composition \[ \displaystyle\sum_{i\in I}V_{i}\otimes W_i\circ\displaystyle\sum_{i'\in I}V'_{i'}\otimes W'_{i'}=\sum_{i\in I}\sum_{i'\in I}V_{i}V'_{i'}\otimes W'_{i'}W_{i}, \] where \(\{V_i,W_i\}_{i\in I}\) is a family in \(\mathcal{B}(\mathcal{H})\) for which each of the series \(\sum_{i\in I}V_iV_{i}^\ast\) and \(\sum_{i\in I}W_iW_{i}^\ast\) is weak\(^\ast\)-convergent.
Using the duality \((\mathcal{V}_{\ast}\otimes^h\mathcal{V}_\ast)^\ast\cong\mathcal{V}\otimes^{w^\ast h}\mathcal{V}\), where \(\mathcal{V}\subseteq\mathcal{B}(\mathcal{H})\) is a von Neumann algebra with predual \(\mathcal{V}_\ast\) and \(\mathcal{V}_{\ast}\otimes^h\mathcal{V}_\ast\) is the completion of \(\mathcal{V}_{\ast}\otimes\mathcal{V}_\ast\) with respect to the norm \(\|u\|_{h}=\displaystyle\sup\{|\langle u,\Phi\rangle|: \Phi\in\mathcal{V}\otimes^{w^\ast h}\mathcal{V},\;\|\Phi\|_{cb}\leq1\}\), and after some details, the authors define the Haagerup tensor product of \(A(G)\) with itself in terms of the completely isometric duality \[ (A(G)\otimes^h A(G))^\ast\cong VN(G)\otimes^{w^\ast h}VN(G)\subset\mathcal{CB}(\mathcal{B(H)}). \] Among the main results, it is shown that the Haagerup tensor product of Fourier algebras behaves exactly as does the operator projective tensor product of Fourier algebras with respect to a defined map \(\Gamma\). Moreover, considering another map \(\check{\Gamma}\) on \(A(G)\otimes^h A(G)\), the authors prove that \(\check{\Gamma}(A(G)\otimes^h A(G))=A(G)\) and \(\check{\Gamma}:A(G)\otimes^h A(G)\rightarrow A(G)\) is a complete quotient map. They notice that the convolution algebra \((A(G),^\ast)\) is completely isomorphic to an operator algebra.