Exotic group C * -algebras of simple Lie groups with real rank one

It is well known that the universal and the reduced group $C^*$-algebra of a locally compact group coincide if and only if the group is amenable. In general, there can be many $C^*$-algebras, called exotic group $C^*$-algebras, which lie between these two algebras. In this article, 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 of $G$. First, we show that the subset of equivalence classes of irreducible unitary $L^{p+}$-representations forms a closed ideal of the unitary dual of the groups under consideration. 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. To this end, it suffices to study the $L^p$-integrability properties of spherical functions of class one representations of $G$. We also obtain partial results for the exceptional Lie group $\mathrm{F}_{4(-20)}$. Our results generalize, with completely 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).