Completely Sidon sets in discrete groups

Abstract

A subset of a discrete group G is called completely Sidon if its span in C * (G) is completely isomorphic to the operator space version of the space ℓ 1 (i.e. ℓ 1 equipped with its maximal operator space structure). We recently proved a generalization to this context of Drury's classical union theorem for Sidon sets: completely Sidon sets are stable under finite unions. We give a different presentation of the proof emphasizing the "interpolation property" analogous to the one Drury discovered. In addition we prove the analogue of the Fatou-Zygmund property: any bounded Hermitian function on a symmetric completely Sidon set Λ ⊂ G \ {1} extends to a positive definite function on G. In the final section, we give a completely isomorphic characterization of the closed span C Λ of a completely Sidon set in C * (G): the dual (in the operator space sense) of C Λ is exact iff Λ is completely Sidon. In particular, Λ is completely Sidon as soon as C Λ is completely isomorphic (by an arbitrary isomorphism) to ℓ 1 (Λ) equipped with its maximal operator space structure.

A Note on Sidon Sets in Bounded Orthonormal Systems

“…[5, p. 28]), it is closely related to Désiré André's reflection principle for Brownian motion (see e.g. [4, p. 558]) and the second one follows from (7). We then set for k ≥ 1 S ′ k = M k∧Tn and…”

“…We end this paper by an outline of a proof that the union of two Sidon sequences is ⊗ 4 -Sidon, more direct than the one in [6]. The route we use avoids the consideration of randomly Sidon sequences, it is essentially the commutative analogue of the proof in [7], with the free Abelian group replacing the free group. The key fact for the latter route is still the following: Lemma 14.…”