Realization of compact spaces as cb-Helson sets

Authors

Yemon Choi

Keywords: Fourier algebra, Helson set, operator space.

MSC 2010: 43A30 (primary), 46L07 (secondary)

DOI: 10.1215/20088752-3429526

Contribution to special issue in honour of A.T.-M. Lau.

Status

Appeared as Ann. Funct. Anal. 7 (2016), no. 1, 158–169.

Preprint version available at arXiv 1504.03597

Reviews

[ Math Review | Zentralblatt (summary) ]

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]. A negative result from the same paper is also improved.

This is a sequel to my 2013 paper with Ebrahim Samei, where we introduced the notion of a cb-Helson subset of a locally compact group. The negative result mentioned in the abstract above was our observation that a virtually abelian group cannot contain an infinite cb-Helson set; and the improved version in the present paper is that if $G$ is a locally compact group whose underlying discrete group is amenable, then $G$ cannot contain an infinite cb-Helson set. In particular, this applies to all locally compact groups that are (virtually) solvable.
The compact groups ${\mathbb G}$ that are constructed in the paper are infinite products of connected compact Lie groups. At time of writing (February 2021) it remains unknown if there are any Lie groups that contain infinite cb-Helson sets.