PROCEEDINGS of the AMERICAN MATHEMATICAL SOCIETY Volume 84, Number 4, April 1982 MULTIPLIERS AND ASYMPTOTIC BEHAVIOUR OF THE FOURIER ALGEBRAOF NONAMENABLE GROUPS CLAUDIO NEBBIA1 Abstract. Let G be a locally compact group and A{G) the algebra of matrix coefficients of the regular representation. We prove that G is amenable if and only if there exist functions u E A(G) which vanish at infinity at any arbitrarily slow rate. The "only if' part of the result was essentially known. With the additional hypothesis that G be discrete, we deduce that G is amenable if and only if every multiplier of the algebra A{G) is a linear combination of positive definite functions. Again, the "only if' part of this result was known. 1. Introduction. Let G be a locally compact group; let CQ(G) be the algebra of continuous complex-valued functions on G which vanish at infinity. We use the definitions and the terminology of [3]. We let B(G) be the Fourier-Stieltjes algebra consisting of all matrix coefficients of unitary representations of G, and A(G) the Fourier algebra consisting of matrix coefficients of the regular representation. Regarded as the dual space of C*(G) (completion of LX(G) in the minimal regular norm), B(G) is a Banach algebra under pointwise multiplication and A(G) is a closed ideal in B(G). We let VN(G) be the von Neumann algebra of the regular representation X of G, which is the dual space of A(G). It is known and easy to prove that if G is amenable there exist functions in A(G) which vanish at infinity at arbitrarily slow rates. More specifically, if G is amenable, for every f e C^G), there exist u e A(G) and g e C0(G) such that f(x) = u(x)g(x). This means of course that/(x) = o(u(x)) as x —» oo. We show that this property is characteristic of amenable groups. Theorem 1 contains this result and other equivalent char- acterizations of amenability. Hence if G is nonamenable, the coefficients of the regular representation must satisfy some condition of decrease at infinity. For particular nonamenable groups specific significant conditions are known; for instance, if G is a semisimple Lie group with finite center, the Kunze-Stein phenomenon [2] implies that A(G) c np>2Lp(G), and if G is a free group with at least two generators, a coefficient u(x) of the regular representation must satisfy the condition {2|x|_„|u(x)|2}1/2 = 0(ri) where |jc| denotes the length of the reduced word x [9, Theorem 3.1, p. 291]. Received by the editors December 4, 1980. 1980 Mathematics Subject Classification. Primary 43A07, 43A22. Key words and phrases. Locally compact group, amenable group, multiplier of the Fourier algebra, Fourier-Stieltjes algebra, asymptotic behaviour of coefficients of the regular representation. 'The contents of this paper are part of the author's thesis for the Laurea in Mathematics at the University of Rome written under the supervision of Professor A. Figà-Talamanca. This research is supported by "Istituto Nazionale di Alta Matemática F. Seven". © 1982 American Mathematical Society 0002-9939/81/0000-1090/S02.50 549 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use