Realization of compact spaces as cb-Helson sets

TL;DR

This paper demonstrates that any compact Hausdorff space can be embedded into a compact group as a cb-Helson set, showing the existence of such sets within compact groups and advancing understanding of their structure.

Contribution

It constructs embeddings of arbitrary compact spaces into compact groups with cb-Helson properties, answering a previously open question and improving known negative results.

Findings

Existence of compact groups containing infinite cb-Helson subsets

Construction of homeomorphic embeddings with complete quotient maps

Extension of previous negative results on cb-Helson sets

Abstract

We show that, given a compact Hausdorff space $\Omega$, there is a compact group ${\mathbb G}$ and a homeomorphic embedding of $\Omega$ into ${\mathbb G}$, such that the restriction map ${\rm A}({\mathbb G})\to C(\Omega)$ is a complete quotient map of operator spaces. In particular, this shows that there exist compact groups which contain infinite cb-Helson subsets, answering a question raised in [Choi--Samei, Proc. AMS 2013; cf. http://arxiv.org/abs/1104.2953]. A negative result from the same paper is also improved.