  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.

Functional Analysis;Operator Algebras