Algebras of diagonal operators of the form scalar-plus-compact are Calkin algebras

Abstract

For every Banach space $X$ with a Schauder basis consider the Banach algebra $\mathbb{R} I\oplus\mathcal{K}_\mathrm{diag}(X)$ of all diagonal operators that are of the form $\lambda I + K$. We prove that $\mathbb{R} I\oplus\mathcal{K}_\mathrm{diag}(X)$ is a Calkin algbra i.e., there exists a Banach space $\mathcal{Y}_X$ so that the Calkin algebra of $\mathcal{Y}_X$ is isomorphic as a Banach algebra to $\mathbb{R} I\oplus\mathcal{K}_\mathrm{diag}(X)$. Among other applications of this theorem we obtain that certain hereditarily indecomposable spaces and the James spaces $J_p$ and their duals endowed with natural multiplications are Calkin algebras, that all non-reflexive Banach spaces with unconditional bases are isomorphic as Banach spaces to Calkin algebras, and that sums of reflexive spaces with unconditional bases with certain James-Tsirelson type spaces are isomorphic as Banach spaces to Calkin algebras.