Embedding C⁎-algebras into the Calkin algebra of ℓp

Let $p\in (1,\infty )$. We show that there is an isomorphism from any separable unital subalgebra of $B\left({\ell }^2\right)/K\left({\ell }^2\right)$ onto a subalgebra of $B\left({\ell }^p\right)/K\left({\ell }^p\right)$ that preserves the Fredholm index. As a consequence, every separable $C^{⁎}$-algebra is isomorphic to a subalgebra of $B\left({\ell }^p\right)/K\left({\ell }^p\right)$. 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.