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

Abstract

Let $p\in (1,\infty )\backslash \{2\}$. We show that every homomorphism from a $C^{\ast }$-algebra $\mathcal{A}$ into $B(l^p(J))$ 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 $B(l^p(J))$, for some set $J$, if and only if $\mathcal{A}$ is residually finite dimensional.