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

Niels Jakob Laustsen¹ ; Richard Skillicorn¹

¹ Department of Mathematics and Statistics Fylde College Lancaster University Lancaster LA1 4YF, United Kingdom

Studia Mathematica, Tome 236 (2017) no. 1, pp. 51-62

Résumé

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.

Détail

DOI : 10.4064/sm8554-9-2016

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