Convolutions on the Haagerup tensor products of Fourier algebras (Q2822580)

Let \(G\) be a compact group and \(A(G)\) be its Fourier algebra as defined by \textit{P. Eymard} [Bull. Soc. Math. Fr. 92, 181--236 (1964; Zbl 0169.46403)]. It was shown in [\textit{E. G. Effros} and \textit{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 \(A(G)\otimes^h A(G)\).NEWLINENEWLINEIf \(\mathcal{H}\) 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 NEWLINE\[NEWLINE\|\Phi\|_{cb}=\min\bigg\{\bigg\|\displaystyle\sum_{i\in I}V_iV_{i}^\ast \bigg\|^{1/2}\bigg\|\displaystyle\sum_{i\in I}W_{i}^\ast W_i\bigg\|^{1/2}: \Phi=\displaystyle\sum_{i\in I}V_{i}\otimes W_{i} \bigg\} NEWLINE\]NEWLINE and operator composition NEWLINE\[NEWLINE\displaystyle\sum_{i\in I}V_{i}\otimes W_i\circ\displaystyle\sum_{i'\in I}V'_{i'}\otimes W'_{i'}=\sum_{i\in I}\sum_{i'\in I}V_{i}V'_{i'}\otimes W'_{i'}W_{i},NEWLINE\]NEWLINE 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.NEWLINENEWLINEUsing 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 NEWLINE\[NEWLINE(A(G)\otimes^h A(G))^\ast\cong VN(G)\otimes^{w^\ast h}VN(G)\subset\mathcal{CB}(\mathcal{B(H)}). NEWLINE\]NEWLINE 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.