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.
Categories: math.FA math.OA