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

TL;DR

The paper demonstrates that any separable unital subalgebra of the Calkin algebra on a Hilbert space can be embedded into the Calkin algebra on an er space , preserving Fredholm indices, thus linking operator algebras across different er spaces.

Contribution

It establishes an isomorphism from any separable unital subalgebra of the Hilbert space Calkin algebra to a subalgebra of the er space Calkin algebra, preserving Fredholm indices.

Findings

Any separable C*-algebra can be embedded into the Calkin algebra of er spaces.

Existence of operators on er spaces with arbitrary Fredholm indices.

Extension of Brown-Douglas-Fillmore theory to er spaces.

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 $C^{*}$-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.