$C^{*}$-algebras isomorphically representable on $l^{p}$

Abstract

Let $p\in \left(1,\infty \right)\setminus \left\{2\right\}$. We show that every homomorphism from a $C^{\ast }$-algebra $\mathcal{𝒜}$ into $B\left(l^p\left(J\right)\right)$ satisfies a compactness property where $J$ is any set. As a consequence, we show that a $C^{\ast }$-algebra $\mathcal{𝒜}$ is isomorphic to a subalgebra of $B\left(l^p\left(J\right)\right)$, for some set $J$, if and only if $\mathcal{𝒜}$ is residually finite-dimensional.