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

TL;DR

This paper extends Read's construction of a Banach space with a discontinuous derivation on its operator algebra, by analyzing the weakly compact operators and the associated algebraic structures.

Contribution

It generalizes Read's main theorem and technical lemmas through the construction of a split-exact sequence involving weakly compact operators and a unitized Hilbert space.

Findings

Constructed a split-exact sequence involving $ ilde{ ext{l}_2}$ and $ ext{W}(E_{ ext{R}})$

Extended Read's theorem to a broader class of Banach spaces

Provided new insights into discontinuous derivations on operator algebras

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 generalise 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} --> $\mathscr{W}(E_{\text{R}})$ --> $\mathscr{B}(E_{\text{R}})$--> $\tilde{\ell_2}$-->{0}, where $\mathscr{W}(E_{\text{R}})$ denotes the ideal of weakly compact operators on $E_{\text{R}}$, while $\tilde{\ell_2}$ is the unitization of the Hilbert space $\ell_2$, endowed with the zero product.