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

## TL;DR

This paper introduces the concept of completely Sidon sets in $C^*$-algebras, generalizing classical Sidon set stability under unions to a non-commutative operator space context, with applications to free group $C^*$-algebras.

## Contribution

It extends Drury's theorem to completely Sidon sets in $C^*$-algebras, including non-commutative generalizations related to free groups and von Neumann algebras.

## Key findings

- Complete Sidon sets are stable under finite unions in $C^*$-algebras.
- The non-commutative version of Drury's theorem is established.
- Extensions to von Neumann algebras with tracial states are provided.

## 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.