  We consider the issue of describing all self-adjoint idempotents
(projections) in $L^1(G)$ when $G$ is a unimodular locally compact group. The
approach is to take advantage of known facts concerning subspaces of the
Fourier-Stieltjes and Fourier algebras of $G$ and the topology of the dual
space of $G$. We obtain an explicit description of any projection in $L^1(G)$
which happens to also lie in the coefficient space of a finite direct sum of
irreducible representations. This leads to a complete description of all
projections in $L^1(G)$ for $G$ belonging to a class of groups that includes
$SL(2,R)$ and all almost connected nilpotent locally compact groups.
