Groups whose Fourier algebra and Rajchman algebra coincide

TL;DR

This paper investigates the class of locally compact groups where the Fourier algebra and Rajchman algebra are identical, revealing structural properties, existence results, and conditions affecting this coincidence.

Contribution

It provides new structural insights, existence results for non-compact groups, and criteria for when Fourier and Rajchman algebras coincide, extending previous work.

Findings

Existence of uncountably many non-compact groups with coinciding algebras

No non-compact nilpotent group has this property

Certain solvable groups do have this property

Abstract

We study locally compact groups for which the Fourier algebra coincides with the Rajchman algebra. In particular, we show that there exist uncountably many non-compact groups with this property. Generalizing a result of Hewitt and Zuckerman, we show that no non-compact nilpotent group has this property, whereas non-compact solvable groups with this property are known to exist. We provide several structural results on groups whose Fourier and Rajchman algebras coincide as well as new criteria for establishing this property. Finally, we study the relation between groups with completely reducible regular representation and groups whose Fourier and Rajchman algebras coincide. For unimodular groups with completely reducible regular representation, we show that the Fourier algebra may in general be strictly smaller than the Rajchman algebra.