## 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

### MSC:

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