Operator Sidon sets

A subset $\Lambda$ of a discrete group $G$ is called "completely Sidon" (or "operator Sidon") if any bounded function $f: \Lambda\to B(H)$ extends to a c.b. map $\tilde f: C^*(G)\to B(H)$. Equivalently, the closed span of $\Lambda$ in $C^*(G)$, denoted by $C_\Lambda$, is completely isomorphic to the operator space version of the space $\ell_1$ (i.e. $\ell_1$ equipped with its maximal operator space structure).  The typical example is a free set. Only non-amenable groups can contain infinite completely Sidon sets.  Such sets have been previously considered by Bożejko.  We generalize to this context Drury's classical theorem: completely Sidon sets are stable under finite unions.  We also obtain the operator valued analogue of the "Fatou-Zygmund property": any bounded $f: \Lambda\to B(H)$ on an asymmetric completely Sidon set extends to a (completely) positive definite function on $G$.  We give a completely isomorphic characterization of completely Sidon sets: $\Lambda $ is completely Sidon iff the operator space $C_\Lambda$ is completely isomorphic (by an arbitrary isomorphism) to $\ell_1(\Lambda)$. This is the operator space version of a result of Varopoulos for classical Sidon sets.  We will also discuss the systems of non-commutative random variables that are "dominated by free-Gaussians", in analogy with the classical subGaussian systems.