  A locally compact group $G$ is said to be $\ast$-regular if the natural map
$\Psi:\Prim C^\ast(G)\to\Prim_{\ast} L^1(G)$ is a homeomorphism with respect to
the Jacobson topologies on the primitive ideal spaces $\Prim C^\ast(G)$ and
$\Prim_{\ast} L^1(G)$. In 1980 J. Boidol characterized the $\ast$-regular ones
among all exponential Lie groups by a purely algebraic condition. In this
article we introduce the notion of $L^1$-determined ideals in order to discuss
the weaker property of primitive $\ast$-regularity. We give two sufficient
criteria for closed ideals $I$ of $C^\ast(G)$ to be $L^1$-determined. Herefrom
we deduce a strategy to prove that a given exponential Lie group is primitive
$\ast$-regular. The author proved in his thesis that all exponential Lie groups
of dimension $\le 7$ have this property. So far no counter-example is known.
Here we discuss the example $G=B_5$, the only critical one in dimension $\le
5$.
