MULTIPLIER NORM OF FINITE SUBSETS OF DISCRETE GROUPS

Let G be a discrete group, VN(G) the von Neumann algebra generated by the left translation operators and A(G) the Fourier algebra of G. Then VN (G) can be considered as a subspace of l2(G) and A(G) = l2(G) ∗ l2(G). The Banach space dual of A(G) is VN(G). The duality between φ ϵ VN(G) and u ϵ A(G) is <φ, u> = ΣxϵG φ(x)u(x), if φ ϵl1(G) or u ϵl2(G); in these two cases, φu ϵl1(G). We will show that if G is infinite then there exist φ ϵ VN (G) and u ϵ A(G) such that φuϵ/l 1(G). When G is countably infinite, this implies that there exist φ ϵ VN(G), u ϵ A(G) and an increasing sequence of finite subsets Fn, G = ∪Fn such that the sequence σxϵFn φ(x)u(x) is divergent and therefore ||XFn||MA(G) is unbounded. This leads us to give a preliminary study of the growth of ||XF||MA(G) as |F|, the size of the finite subset F of G, is increasing. Recall that when G = Z, the additive group of integers, ||XF||MA(Z) = 1/2π∫2π0|ΣkϵFeikx|dx.