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

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