for a non-amenable group $g$, there may be many (exotic) group $c^*$-algebras that lie naturally between the universal and the reduced $c^*$-algebra of $g$. let $g$ be a simple lie group or an appropriate locally compact group acting on a tree. i will explain how the $l^p$-integrability properties of different spherical functions on $g$ (relative to a maximal compact subgroup) can be used to distinguish between different (exotic) group $c^*$-algebras. this recovers results of samei and wiersma. additionally, i will explain that under certain natural assumptions, the aforementioned exotic group $c^*$-algebras are the only ones coming from $g$-invariant ideals in the fourier-stieltjes algebra of $g$. \\ \\ this is based on joint work with dennis heinig and timo siebenand.