Embedding C*-algebras into the Calkin algebra of $\ell^{p}$

Let $p\in(1,\infty)$. We show that there is an isomorphism from any separable unital subalgebra of $B(\ell^{2})/K(\ell^{2})$ onto a subalgebra of $B(\ell^{p})/K(\ell^{p})$ that preserves the Fredholm index. As a consequence, every separable $C^{*}$-algebra is isomorphic to a subalgebra of $B(\ell^{p})/K(\ell^{p})$. Another consequence is the existence of operators on $\ell^{p}$ that behave like the essentially normal operators with arbitrary Fredholm indices in the Brown-Douglas-Fillmore theory.