  Exotic group $C^*$-algebras are $C^*$-algebras that lie between the universal
and the reduced group $C^*$-algebra of a locally compact group. We consider
simple Lie groups $G$ with real rank one and investigate their exotic group
$C^{*}$-algebras $C^*_{L^{p+}}(G)$, which are defined through
$L^p$-integrability properties of matrix coefficients of unitary
representations. First, we show that the subset of equivalence classes of
irreducible unitary $L^{p+}$-representations forms a closed ideal of the
unitary dual of these groups. This result holds more generally for groups with
the Kunze-Stein property. Second, for every classical simple Lie group $G$ with
real rank one and every $2 \leq q < p \leq \infty$, we determine whether the
canonical quotient map $C^*_{L^{p+}}(G) \twoheadrightarrow C^*_{L^{q+}}(G)$ has
non-trivial kernel. Our results generalize, with different methods, recent
results of Samei and Wiersma on exotic group $C^*$-algebras of
$\mathrm{SO}_{0}(n,1)$ and $\mathrm{SU}(n,1)$. In particular, our approach also
works for groups with property (T).
