Multipliers on Modules Over the Fourier Algebra

Let G be an infinite compact group and G its dual. For 1< p < oom P(G) is a module over 21(G) A(G), the Fourier algebra of G. For 1 < p, q < o0, let Ip q = HomA(G)( P(G), ?'(G)). If G is abelian, then Jll is the space of LP(G)-multipliers. For 1 < p < 2 and p the conjugate index of p, A(G) -1,1 p,p p, p 2, 2 (G). Further, the space ll is the dual of a space called d p a subspace of eo(G). Using a methoa of J. F. Price we observe that U I q, q: I < q < p} NflPI p C=.n0Rq, q: p < q < 2} (where 1 < p < 2). Finally, S11 =10} for 1 < p < q < oo. q, p 1. Modules, over the Fourier algebra. Throughout this paper G will denote an infinite compact group and G its dual (we use the notation from [1]). Throughout, 1 < p, q, r < oo. Given p, the conjugate index will be denoted by p' (i/p + 1/p' = 1). Definition. Let b e eF(G) and so += f for / a trigonometric polynomial on G. We define b by the rule / = (f) where f(x) = /(x-l), x E G. Proposition 1. The map b b from CF (6) to eF(G) extends to an isometry of SP (G) (1 < p < 00) and of E0G) Proof. For f a trigonometric polynomial on G, we have that (/)= ((/)*)= ((/) )* (J/(J)* (see [1, p. 87]). Thus for b E CF(G), oIIlp = IkIlp. El Definition. Let k F, eiwe define E x q Fe CF by the rule x qf) = Oq (0 denotes the inverse Fourier transform of b [1, p. 97]). We note that |kbx ||1 ? | I IR EF(G) (see [1, p. 93]). We define the pairing (s, i/) = Tr(oGr) = (6 * ( /)Y)(e) = fG 7(x)\(x)dmc(x), (k, qi E F(G), Presented to the Society, October 18, 1971; received by the editors November 3, 1971. AMS (MOS) subject classifications (1970). Primary 43A15, 43A22; Secondary 46E30, 46L20.