A subset $\Lambda\subset G$ in a discrete group $G$ is called completely Sidon if every bounded function $\varphi: \Lambda \to B(H)$ on $\Lambda$ extends (linearly) to a c.b. map from $C^*(G)$ to $B(H)$. In other words the closed span of $\Lambda$ in $C^*(G)$ is completely isomorphic to $\ell_1(\Lambda)$ (equipped with its maximal operator space structure). This is the operator space analogue of the notion of Sidon set for Abelian groups (in which case $B(H)=\C$). However, only non-amenable groups (e.g. free groups) can contain infinite completely Sidon sets. In analogy with Drury's theorem for Abelian groups, we show that completely Sidon sets are stable under finite unions. Various generalizations to subsets of $C^*$-algebras will be discussed.
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