A singular, admissible extension which splits algebraically, but not strongly, of the algebra of bounded operators on a Banach space

TL;DR

This paper constructs a specific Banach algebra extension of the algebra of bounded operators on a special Banach space, which splits algebraically but not strongly, answering a longstanding open question.

Contribution

It demonstrates the existence of a singular, admissible extension of the algebra of bounded operators that splits algebraically but not strongly, resolving a question from prior research.

Findings

Existence of a singular, admissible extension that splits algebraically

Extension does not split strongly, showing a separation between algebraic and strong splitting

Addresses a question from Bade, Dales, and Lykova's work

Abstract

Let $E$ be the Banach space constructed by Read (J. London Math. Soc. 1989) such that the Banach algebra $\mathscr{B}(E)$ of bounded operators on $E$ admits a discontinuous derivation. We show that $\mathscr{B}(E)$ has a singular, admissible extension which splits algebraically, but does not split strongly. This answers a natural question going back to the work of Bade, Dales, and Lykova (Mem. Amer. Math. Soc. 1999), and complements recent results of Laustsen and Skillicorn (C. R. Math. Acad. Sci. Paris, to appear).