The compact operators on $c_0$ as a Calkin algebra

Abstract

For a Banach space $X$, let $\mathcal{L}(X)$ denote the algebra of all bounded linear operators on $X$ and let $\mathcal{K}(X)$ denote the compact operator ideal in $\mathcal{L}(X)$. The quotient algebra $\mathcal{L}(X)/\mathcal{K}(X)$ is called the Calkin algebra of $X$, and it is denoted $\mathcal{C}al(X)$. We prove that the unitization of $\mathcal{K}(c_0)$ is isomorphic as a Banach algebra to the Calkin algebra of some Banach space ${\mathcal{Z}}_{\mathcal{K}(c_0)}$. This Banach space is an Argyros-Haydon sum $({\oplus }_{n=1}^{\infty }{X}_n{)}_{\mathrm{AH}}$ of a sequence of copies $X_n$ of a single Argyros-Haydon space ${\mathfrak{X}}_{\mathrm{AH}}$, and the external versus the internal Argyros-Haydon construction parameters are chosen from disjoint sets.

23 pages, no figures