Michigan Mathematical Journal

Algebras of Diagonal Operators of the Form Scalar-Plus-Compact Are Calkin Algebras

Pavlos Motakis, Daniele Puglisi, and Andreas Tolias

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Abstract

For every Banach space X with a Schauder basis, consider the Banach algebra RIKdiag(X) of all diagonal operators that are of the form λI+K. We prove that RIKdiag(X) is a Calkin algebra, that is, there exists a Banach space YX such that the Calkin algebra of YX is isomorphic as a Banach algebra to RIKdiag(X). Among other applications of this theorem, we obtain that certain hereditarily indecomposable spaces and the James spaces Jp 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
Received: 15 January 2018
Revised: 8 February 2018
First available in Project Euclid: 27 November 2019

Permanent link to this document
https://projecteuclid.org/euclid.mmj/1574845272

Digital Object Identifier
doi:10.1307/mmj/1574845272

Mathematical Reviews number (MathSciNet)
MR4071347

Zentralblatt MATH identifier
07208927

Subjects
Primary: 46B03: Isomorphic theory (including renorming) of Banach spaces 46B25: Classical Banach spaces in the general theory 46B28: Spaces of operators; tensor products; approximation properties [See also 46A32, 46M05, 47L05, 47L20] 46B45: Banach sequence spaces [See also 46A45]

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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