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

Abstract

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 $C^{\ast }$-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.

To appear in Journal of Functional Analysis