## Michigan Mathematical Journal

### 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 algebra, that is, there exists a Banach space $\mathcal{Y}_{X}$ such 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 nonreflexive 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.

#### Article information

Source
Michigan Math. J., Volume 69, Issue 1 (2020), 97-152.

Dates
Revised: 8 February 2018
First available in Project Euclid: 27 November 2019

https://projecteuclid.org/euclid.mmj/1574845272

Digital Object Identifier
doi:10.1307/mmj/1574845272

Mathematical Reviews number (MathSciNet)
MR4071347

Zentralblatt MATH identifier
07208927

#### Citation

Motakis, Pavlos; Puglisi, Daniele; Tolias, Andreas. Algebras of Diagonal Operators of the Form Scalar-Plus-Compact Are Calkin Algebras. Michigan Math. J. 69 (2020), no. 1, 97--152. doi:10.1307/mmj/1574845272. https://projecteuclid.org/euclid.mmj/1574845272

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