Interpolation and Fatou-Zygmund property for completely Sidon subsets of discrete groups (New title: Completely Sidon sets in discrete groups)

Abstract

A subset of a discrete group $G$ is called completely Sidon if its span in $C^{\ast }(G)$ is completely isomorphic to the operator space version of the space ${\ell }_1$ (i.e. ${\ell }_1$ 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 $\Lambda \subset G\setminus \{1\}$ extends to a positive definite function on $G$. In the final section, we give a completely isomorphic characterization of the closed span $C_{\Lambda }$ of a completely Sidon set in $C^{\ast }(G)$: the dual (in the operator space sense) of $C_{\Lambda }$ is exact iff $\Lambda$ is completely Sidon. In particular, $\Lambda$ is completely Sidon as soon as $C_{\Lambda }$ is completely isomorphic (by an arbitrary isomorphism) to ${\ell }_1(\Lambda )$ equipped with its maximal operator space structure.

In v2 we show that if the closed span of a subset of G in C*(G) is completely isomorphic to ${\ell }_1$ (by an arbitrary isomorphism) or if the dual operator space is exact then the set is completely Sidon. v3: more polished, longer with a new title. v4: some new paragraphs taken from another arxiv paper by same author will create a detected overlap. More open problems stated

References in corpus (2)

• A simple proof of a theorem of Kirchberg and related results on $C^{\ast }$-norms
• Multipliers and lacunary sets in non-amenable groups