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 $\mathrm{A}\left(\mathbb{G}\right)\to C(\Omega )$ 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 by Choi and Samei. A negative result from that paper is also improved.