. FA ] 9 D ec 1 99 2 MULTIPLIERS AND LACUNARY SETS IN NON - AMENABLE GROUPS

C is called a completely bounded multiplier (= Herz-Schur multiplier) if the transformation defined on the linear span K(G) of {λ(x), x ∈ G} by x∈G f (x)λ(x) → x∈G f (x)ϕ(x)λ(x) is completely bounded (in short c.b.) on the C *-algebra C * λ (G) which is generated by λ (C * λ (G) is the closure of K(G) in B(ℓ 2 (G), ℓ 2 (G)).) One of our main results (stated below as Theorem 0.1) gives a simple characterization of the functions ϕ such that εϕ is a c.b. multiplier on C * λ (G) for any bounded function ε, or equivalently for any choice of signs ε(x) = ±1. We wish to consider also the case when this holds for " almost all " choices of signs. To make this precise, equip {−1, 1} G with the usual uniform probability measure. We will say that εϕ is a c.b. multiplier of C * λ (G) for almost all choice of signs ε if there is a measurable subset Ω ⊂ {−1, 1} G of full measure (note that Ω depends only on countably many coordinates) such that for any ε in Ω εϕ is a c.b. multiplier of C * λ (G). (Note that ϕ is necessarily countably supported when this holds, so the measurability issues are irrelevant.)