Completely Sidon sets in $C^*$-algebras (New title)

Abstract

A sequence in a $C^*$-algebra $A$ is called completely Sidon if its span in $A$ 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 latter can also be described as the span of the free unitary generators in the (full) $C^*$-algebra of the free group $\F_\infty$ with countably infinitely many generators. Our main result is a generalization to this context of Drury's classical theorem stating that Sidon sets are stable under finite unions. In the particular case when $A=C^*(G)$ the (maximal) $C^*$-algebra of a discrete group $G$, we recover the non-commutative (operator space) version of Drury's theorem that we recently proved. We also give several non-commutative generalizations of our recent work on uniformly bounded orthonormal systems to the case of von Neumann algebras equipped with normal faithful tracial states.