a unitary representation $\pi\colon g\to b(h)$ of a locally compact group $g$ is an \emph{$l^p$-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 \emph{$l^p$-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\textnormal{ is an }l^p\textnormal{-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 $2\leq p' \leq p \leq \infty$ 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.