Completely co-bounded Schur multipliers (Q2912196)

Let \(E\) and \(F\) be operator spaces, and \(F^{op}\) be the operator space opposite to \(F\). A linear map \(u : E\to F\) is called completely co-bounded if it is completely bounded as a map from \(E\) into \(F^{op}\); in this case, \(\|u\|_{cob}\) denotes the completely bounded norm of \(u : E\to F^{op}\).NEWLINENEWLINELet \(\varphi = (\varphi_{i,j})_{i,j\in \mathbb{N}}\) be a Schur multiplier and let \(M_{\varphi} : B(\ell_2) \to B(\ell_2)\) be the map of Schur multiplication by \(\varphi\). It is shown that \(M_{\varphi}\) is completely co-bounded if and only if the matrix \((|\varphi_{i,j}|)_{i,j\in \mathbb{N}}\) defines a bounded operator on \(\ell_2\). Moreover, in this case, \(\|M_{\varphi}\|_{cob} = \|(|\varphi_{i,j}|)_{i,j\in \mathbb{N}}\|_{B(\ell_2)}\). More generally, completely co-bounded Schur multipliers on the Schatten class \(\mathcal{S}_p\) are characterised, in terms of completely bounded factorisation through \(\ell_p(\mathbb{N}\times\mathbb{N})\), \(2\leq p\leq \infty\).NEWLINENEWLINECompletely co-bounded Herz-Schur multipliers are also studied; in particular, the identity map on the \(C^*\)-algebra \(C^*(G)\) of a finite group \(G\) is shown to have completely co-bounded norm equal to \(\sup_{\pi\in \hat{G}} \dim(\pi)\), where, as usual, \(\hat{G}\) denotes a maximal set of pairwise inequivalent irreducible representations of \(G\) and \(\dim(\pi)\) is the dimension of the Hilbert space on which \(\pi\) acts. The latter result rests on the fact, also established in this paper, that the completely co-bounded norm of the map \(u : B(H)\to B(H)\) (where \(H\) is a Hilbert space) given by \(u(x) = axb\) for some \(a,b\in B(H)\), is equal to \(\|a\|_2\|b\|_2\), where \(\|c\|_2\) is the Hilbert-Schmidt norm of \(c\) (set to be equal to infinity if \(c\) does not belong to \(\mathcal{S}_2\)). Unlike the completely bounded case, the completely co-bounded norm of a Herz-Schur multiplier on the reduced \(C^*\)-algebra of a finite group \(G\), arising from a function \(f : G\to \mathbb{C}\), is shown to differ from the completely co-bounded norm of \(M_{N(f)}\), where \(N(f)(s,t) = f(st^{-1})\), \(s,t\in G\).