  Let $G$ be an exponential solvable Lie group. By definition $G$ is
$\ast$-regular if $ker_{L^1(G)}\pi$ is dense in $ker_{C^\ast(G)}\pi$ for all
unitary representations $\pi$ of $G$. Boidol characterized the $\ast$-regular
exponential Lie groups by a purely algebraic condition. In this article we will
focus on non-$\ast$-regular groups. We say that $G$ is primitive $\ast$-regular
if the above density condition is satisfied for all irreducible
representations. Our goal is to develop appropriate tools to verify this weaker
property. To this end we will introduce Duflo pairs $(W,p)$ and central Fourier
multipliers $\psi$ on the stabilizer $M=G_fN$ of representations $\pi=\mcK(f)$
in general position. Using Littlewood-Paley theory we will derive some results
on multiplier operators $T_\psi$ which might be of independent interest. The
scope of our method of separating triples $(W,p,\psi)$ will be sketched by
studying two significant examples in detail. It should be noticed that the
methods 'separating triples' and 'restriction to subquotients' suffice to prove
that all exponential solvable Lie groups of dimension $\le 7$ are primitive
$\ast$-regular.
