  We consider a new class of potentially exotic group C*-algebras
$C^*_{PF_p^*}(G)$ for a locally compact group $G$, and its connection with the
class of potentially exotic group C*-algebras $C^*_{L^p}(G)$ introduced by
Brown and Guentner. Surprisingly, these two classes of C*-algebras are
intimately related. By exploiting this connection, we show
$C^*_{L^p}(G)=C^*_{PF_p^*}(G)$ for $p\in (2,\infty)$, and the C*-algebras
$C^*_{L^p}(G)$ are pairwise distinct for $p\in (2,\infty)$ when $G$ belongs to
a large class of nonamenable groups possessing the Haagerup property and either
the rapid decay property or Kunze-Stein phenomenon by characterizing the
positive definite functions that extend to positive linear functionals of
$C^*_{L^p}(G)$ and $C^*_{PF_p^*}(G)$. This greatly generalizes earlier results
of Okayasu and the second author on the pairwise distinctness of $C^*_{L^p}(G)$
for $2<p<\infty$ when $G$ is either a noncommutative free group or the group
$SL(2,\mathbb R)$, respectively.
  As a byproduct of our techniques, we present two applications to the theory
of unitary representations of a locally compact group $G$. Firstly, we give a
short proof of the well-known Cowling-Haagerup-Howe Theorem which presents
sufficient condition implying the weak containment of a cyclic unitary
representation of $G$ in the left regular representation of $G$. Also we give a
near solution to a 1978 conjecture of Cowling. This conjecture of Cowling
states if $G$ is a Kunze-Stein group and $\pi$ is a unitary representation of
$G$ with cyclic vector $\xi$ such that the map $$G\ni s\mapsto \langle
\pi(s)\xi,\xi\rangle$$ belongs to $L^p(G)$ for some $2< p <\infty$, then
$A_\pi\subseteq L^p(G)$. We show $B_\pi\subseteq L^{p+\epsilon}(G)$ for every
$\epsilon>0$ (recall $A_\pi\subseteq B_\pi$).
