Realization of compact spaces as 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 $\mathrm{A}(\mathbb{G})\to C(\Omega )$ 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.

v2: AMS-LaTeX, 12 pages. Changes to v1: material on continuity of product representations has been streamlined; other minor changes/corrections made, following referee's recommendations. Accepted by Ann. Funct. Anal; this is, modulo formatting, the author-accepted manuscript (CC-BY licence)