This is an announcement for the paper “Bounded representations on $l^p$” by March T. Boedihardjohttps://arxiv.org/search/math?searchtype=author&query=Boedihardjo%2C+M+T.
Abstract: We show that (i) every bounded unital representation of an amenable group $G$ on $l^{p}$, $1<p<\infty$, is a direct summand of a representation that is approximately similar to the left regular representation of $G$ on $l^{p}$ and that (ii) if $\rho$ is a unital representation of a unital $C^{*}$-algebra $\mathcal{A}$ on $l^{p}$, $1<p<\infty$, $p\neq 2$, then $\rho$ satisfies a compactness property and $\mathcal{A}/\text{ker }\rho$ is residually finite dimensional. As a consequence, a separable unital $C^{*}$-algebra $\mathcal{A}$ is isomorphic to a subalgebra of $B(l^{p})$, $1<p<\infty$, $p\neq 2$, if and only if $\mathcal{A}$ is residually finite dimensional.
The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1812.11165