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

TL;DR

This paper demonstrates that certain diagonal operator algebras, specifically scalar-plus-compact forms on Banach spaces with Schauder bases, are isomorphic to Calkin algebras, revealing new structural insights into Banach space operator algebras.

Contribution

It proves that these diagonal operator algebras are Calkin algebras, establishing a new link between Banach space operators and Calkin algebra structures.

Findings

Certain hereditarily indecomposable spaces are Calkin algebras

James spaces and their duals are Calkin algebras

Non-reflexive Banach spaces with unconditional bases are isomorphic to 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.