Completely bounded Fourier multipliers over compact groups (Q2910519)

Let \(H\) be a complex Hilbert space and let \(B(H)\) denote the set of bounded operators on \(H\). An operator space is a closed subspace of \(B(H)\). Let \(G\) be a compact group with normalized Haar measure \(\lambda\) and unitary dual \(\Sigma\), and let \(E\) be an operator space. For an equivalence class \(\sigma\) belonging to \(\Sigma\), choose a \(U^{\sigma}\) and write \(H_{\sigma}\) for its hilbertian representation space. The Fourier-Stieltjes transform of a bounded vector measure \(m : G \to E\), which can be interpreted as a family \((\hat{m}(\sigma))_{\sigma \in \Sigma}\) of continuous sesquilinear mappings from \(H_{\sigma} \times H_{\sigma}\) into \(E\), is defined by NEWLINE\[NEWLINE \hat{m}(\sigma) (\xi, \eta) = \int_G \langle \overline{U}_t^{\sigma} \xi, \eta \rangle dm(t), (\xi, \eta) \in H_{\sigma} \times H_{\sigma}. NEWLINE\]NEWLINE The Fourier transform \(\hat{f}\) of a Haar-integrable function \(f : G \to E\) can now be defined by identifying \(f\) with the vector measure \(f \lambda\).NEWLINENEWLINELet \(L_p(G)\) be the \(L_p\)-space of complex-valued functions on \(G\), and the \(L_p\)-space \(E\)-valued functions will be indicated by \(L_p(G, E)\). Let \(\varphi : \Sigma \to \mathbb{C}\) be a map. The Fourier multiplier \(M_{\varphi} f\) is defined on both \(L_p(G, E)\) and \(L_p(G)\), where \(\hat{f}\) has finite support. The function \(\varphi\) is said to be a bounded multiplier on \(L_p(G)\) if the map \(M_{\varphi}\) is bounded on \(L_p(G)\). Furthermore, \(\varphi\) is defined to be a completely bounded multiplier on \(L_p(G)\) if \(M_{\varphi}\) is completely bounded on \(L_p(G)\). One of the main results of the paper under review is: \(M_{\varphi}\) is a Fourier multiplier on \(L_p(G, E)\) if and only if for all \(f \in L_p(G, E), \widehat{M_{\varphi}f} = \varphi \hat{f}\). The other main result of this paper gives a characterization of completely bounded multipliers on \(L_p(G)\).