Extensions and the weak Calkin algebra of Read’s Banach space admitting discontinuous derivations
Volume 236 / 2017
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.
