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.

49 pages