MULTIPLIERS AND LACUNARY SETS IN NON-AMENABLE GROUPS
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abstract
Let $G$ be a discrete group. Let $lambda : G o B(ell_2(G),ell_2(G))$ be the left regular representation. A function $ph : G o comp$ is called a completely bounded multiplier (= Herz-Schur multiplier) if the transformation defined on the linear span $K(G)$ of ${lambda(x),x in G}$ by $$sum_{x in G} f(x) lambda(x) o sum_{x in G} f(x) ph(x) lambda(x)$$ is completely bounded (in short c.b.) on the $C^*$-algebra $C_lambda^*(G)$ which is generated by $lambda$ ($C_lambda^*(G)$ is the closure of $K(G)$ in $B(ell_2(G),ell_2(G))$.) One of our main results gives a simple characterization of the functions $ph$ such that $eps ph$ is a c.b. multiplier on $C_lambda^*(G)$ for any bounded function $eps$, or equivalently for any choice of signs $eps(x) = pm 1$. We also consider the case when this holds for ``almost all" choices of signs.
