We consider a new class of potentially exotic group C*-algebras C⁎(PFp⁎(G)) for a locally compact group G, and its connection with the class of potentially exotic group C*-algebras CLp⁎(G) introduced by Brown and Guentner. Surprisingly, these two classes of C*-algebras are intimately related. By exploiting this connection, we show CLp⁎(G)=C⁎(PFp⁎(G)) for p∈(2,∞), and the C*-algebras CLp⁎(G) are pairwise distinct for p∈(2,∞) 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 CLp⁎(G) and C⁎(PFp⁎(G)). This greatly generalizes earlier results of Okayasu (see [30]) and the second author (see [40]) on the pairwise distinctness of CLp⁎(G) for 2<p<∞ when G is either a noncommutative free group or the group SL(2,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 (see [14]). Also we give a near solution to a 1978 conjecture of Cowling stated in [10]. This conjecture of Cowling states if G is a Kunze-Stein group and π is a unitary representation of G with cyclic vector ξ such that the mapG∋s↦〈π(s)ξ,ξ〉 belongs to Lp(G) for some 2<p<∞, then Aπ⊆Lp(G). We show Bπ⊆Lp+ϵ(G) for every ϵ>0 (recall Aπ⊆Bπ).