Herz-Schur multipliers and completely bounded multipliers of the Fourier algebra of a locally compact group. (English) Zbl 0564.43004

Let G be a locally compact group. The space \(B_ 2(G)\) of Herz-Schur multipliers is the set of functions \(\phi\) defined on G such that the function \(M_{\phi}(x,y)=\phi (x^{-1}y)\) is a pointwise multiplier of the projective tensor product \(L^ 2(G)\otimes L^ 2(G)\). The map \(u\otimes v\to v\cdot u^ v\) embeds \(B_ 2(G)\) into the space of multipliers of the Fourier algebra \(A(G)=L^ 2(G)\cdot L^ 2(G).\)
On the other hand, the multiplier \(M_{\phi}\) of A(G) gives rise, by duality, to a multiplier \(M_{\phi}\) of the von Neumann algebra VN(G). Let \(M_ n\) be the space of \(n\times n\) matrices, and lift \(M_{\phi}\) to a map \(\psi_ n\) defined on \(VN(G)\times M_ n\) by \(\psi_ n(T\times A)=({\mathfrak M}_{\phi}T)\times A.\) Then \(\phi\) is called a completely bounded multiplier of A(G), if \(\sup_{n>0}\| \psi \|_ n<\infty\). The space of completely bounded multipliers, denoted by \(M_ 0A(G)\), was recently introduced and studied by J. de Cannière and U. Haagerup.
This paper shows that \(B_ 2(G)\) is isometrically isomorphic with \(M_ 0A(G)\). The proof of the inclusion \(B_ 2(G)\hookrightarrow M_ 0A(G)\) makes use of results of J. de Cannière and U. Haagerup and the following characterization of \(B_ 2(G)\), due to J. Gilbert: \(\phi \in B_ 2(G)\) if and only if there exist a Hilbert space H and two bounded operators \(P:L^ 1(G)\to H\), \(Q:L^ 1(G)\to H^*\) such that \(\int_{G}\int_{G}\phi (yx^{-1})f(x)(y)dx\quad dy=<P(f),\quad Q(g)>\), \(\forall f,g\in L^ 1\), and \(\| \phi \|_{B_ 2}=\inf \{ \| P\|,\| Q\| \}\), where the infimum is taken over all possible representations of this type.
Reviewer: M.Picardello


43A22 Homomorphisms and multipliers of function spaces on groups, semigroups, etc.
43A35 Positive definite functions on groups, semigroups, etc.