described in [5]. In particular, we note that V 1 ðX; Þ obtains a natural operator space structure. Subordinate to the theory of measurable Schur multipliers, we develop a theory of 'continuous' Schur multipliers, which is based on the theory of the extended Haagerup tensor product of Eros and Ruan in [16] (see also [13]).Letting m now denote the left Haar measure on our locally compact group G, we let V 1 ðGÞ ¼ V 1 ðG; mÞ. We dene the 'invariant part', V 1 inv ðGÞ, of V 1 ðGÞ. Our space V 1 inv ðGÞ is essentially the space VðGÞ of [23]. Our main result is that V 1 inv ðGÞ ffi M cb AðGÞ completely isometrically. Thus we obtain a natural extension of the module action of M cb AðGÞ on VNðGÞ to BðL 2 ðGÞÞ. This is the content of x 5. (It should be noted that the isometric identication V 1 inv ðGÞ ffi M cb AðGÞ has been previously known to Haagerup [20]. This fact was discovered by the author after the present article was completed. However, it is not clear how the methods of that article can be adapted to give a complete isometry.)As an immediate application of our results, we have a systematic approach to determining the functorial properties of M cb AðGÞ. In addition, we can identify some circumstances in which a subspace of the Fourier --Stieltjes algebra, BðGÞ, acts completely isometrically as multipliers of AðGÞ. Another application we obtain is a concrete description of the predual of M cb AðGÞ, QðGÞ, whose existence is recognised in [8]. In the case where G is discrete, Pisier [38] implicitly uses the structure of QðGÞ in his partial solution to the similarity problem for representations of groups. These applications are studied in x 6.We note that there are further applications of our results and methods which are not explored in this note. As an example, the author and L. Turowska, in [43], use some of the methods of the current note to show that if G is compact, then AðGÞ naturally imbeds into CðGÞ CðGÞ (projective tensor product of the continuous functions on G with itself). This is used to generalise, to arbitrary compact groups, results on parallel spectral synthesis due to Varopoulos [48] for compact Abelian groups; and to generalise some results on operator synthesis due to Froelich [18].We should also point out that if G is Abelian, there are related results to ours published by St(rmer [45]. In that article it is shown that there is an isometric representation of the measure algebra, MðGÞ, in B ðL 2 ðGÞÞ, which extends convolution on L 1 ðGÞ. This is generalised to arbitrary locally compact groups by Ghahramani [19]. Neufang [33] has shown that this isometric representation has its range in CB ðL 2 ðGÞÞ. Some of the connections between these results and those of the present article are being explored by Neufang, Ruan and the author [34].Sections 2 and 4 are included to support the exposition of this article, but include results of independant interest. In x 2, a uniform approach is developed for dening the weak* and extended Haagerup tensor products. In x 4 some basic results about completely bound...

For locally compact groups G and H let A(G) denote the Fourier algebra of G and B(H ) the Fourier-Stieltjes algebra of H. Any continuous piecewise affine mapan element of the open coset ring) induces a completely bounded homomorphism : A(G) → B(H ) by setting u = u • on Y and u = 0 off of Y. We show that if G is amenable then any completely bounded homomorphism : A(G) → B(H ) is of this form; and this theorem fails if G contains a discrete nonabelian free group. Our result generalises results of Cohen (Amer.

Abstract. Let G be a locally compact group. It is shown that there exists a natural completely isometric representation of the completely bounded Fourier multiplier algebra M cb A(G), which is dual to the representation of the measure algebra) are intrinsically characterized, and some commutant theorems are proved. It is also shown that for any amenable group G, there is a natural completely isometric representation of UCB(Ĝ) * on B(L 2 (G)), which can be regarded as a duality result of Neufang's completely isometric representation theorem for LU C(G) * .

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Let G be a compact group and C(G) be the C * -algebra of continuous complex-valued functions on G. The paper constructs an imbedding of the Fourier algebra A(G) of G into the algebra V(G) = C(G) ⊗ h C(G) (Haagerup tensor product) and deduces results about parallel spectral synthesis, generalizing a result of Varopoulos. It then characterizes which diagonal sets in G × G are sets of operator synthesis with respect to the Haar measure, using the definition of operator synthesis due to Arveson. This result is applied to obtain an analogue of a result of Froelich: a tensor formula for the algebras associated with the pre-orders defined by closed unital subsemigroups of G.