A unitary representation $\pi\colon G\to B(H)$ of a locally compact group $G$ is an $L^p$-representation if $H$ admits a dense subspace $H_0$ so that the matrix coefficient
$$G\ni s\mapsto \langle \pi(s)\xi,\xi\rangle$$
belongs to $L^p(G)$ for all $\xi\in H_0$. The $L^p$-C*-algebra $C^*_{L^p}(G)$ is the C*-completion $L^1(G)$ with respect to the C*-norm
$$\|f\|_{C^*_{L^p}}:=\sup\{\|\pi(f)\| : \pi\text{ is an }L^p\text{-representation of $G$}\}\qquad (f\in L^1(G)).$$
Surprisingly, the C*-algebra $C^*_{L^p}(G)$ is intimately related to the enveloping C*-algebra of the Banach $*$-algebra $PF^*_p(G)$ ($2\leq p\leq \infty$). Consequently, we characterize the states of $C^*_{L^p}(G)$ as corresponding to positive definite functions that `"almost'' belong to $L^p(G)$ in some suitable sense for "many'' $G$ possessing the Haagerup property, and either the rapid decay property or Kunze-Stein phenomenon. It follows that the canonical map
$$C^*_{L^p}(G)\to C^*_{L^{p'}}(G)$$
is not injective for $p>p'\geq 2$ when $G$ is non-amenable and belongs to the class of groups mentioned above. As a byproduct of our techniques, we give a near solution to a 1978 conjecture of Cowling.

This is primarily based on joint work with E. Samei.