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Extensions and the weak Calkin algebra of Read’s Banach space admitting discontinuous derivations

Volume 236 / 2017

Niels Jakob Laustsen, Richard Skillicorn Studia Mathematica 236 (2017), 51-62 MSC: Primary 46H10, 46M18, 47L10; Secondary 16S70. DOI: 10.4064/sm8554-9-2016 Published online: 10 November 2016

Abstract

Read produced the first example of a Banach space $E_{\text{R}}$ such that the associated Banach algebra $\mathscr{B}(E_{\text{R}})$ of bounded operators admits a discontinuous derivation (J. London Math. Soc., 1989). We generalize Read’s main theorem about $\mathscr{B}(E_{\text{R}})$ from which he deduced this conclusion, as well as the key technical lemmas that his proof relied on, by constructing a strongly split-exact sequence \[ \{0\}\rightarrow\mathscr{W}(E_{\text{R}}) \rightarrow\mathscr{B}(E_{\text{R}}) \leftrightarrows \ell_2^\sim\rightarrow\{0\}, \] where $\mathscr{W}(E_{\text{R}})$ denotes the ideal of weakly compact operators on $E_{\text{R}}$, while $\ell_2^\sim$ is the unitization of the Hilbert space $\ell_2$, endowed with the zero product.

Authors

  • Niels Jakob LaustsenDepartment of Mathematics and Statistics
    Fylde College
    Lancaster University
    Lancaster LA1 4YF, United Kingdom
    e-mail
  • Richard SkillicornDepartment of Mathematics and Statistics
    Fylde College
    Lancaster University
    Lancaster LA1 4YF, United Kingdom
    e-mail

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