Splittings of extensions and homological bidimension of the algebra of bounded operators on a Banach space

TL;DR

This paper constructs a Banach space where the algebra of bounded operators exhibits a singular extension that splits algebraically but not strongly, and the algebra's homological bidimension is at least two, answering open questions.

Contribution

It provides a specific Banach space example demonstrating a singular extension with unique splitting properties and establishes a lower bound for the homological bidimension of its operator algebra.

Findings

Existence of a Banach space with a singular extension that splits algebraically but not strongly.

Homological bidimension of the algebra of bounded operators is at least two.

Answers to open problems posed by Bade, Dales, Lykova, and Helemskii.

Abstract

We show that there exists a Banach space $E$ with the following properties: the Banach algebra $\mathscr{B}(E)$ of bounded, linear operators on $E$ has a singular extension which splits algebraically, but it does not split strongly, and the homological bidimension of $\mathscr{B}(E)$ is at least two. The first of these conclusions solves a natural problem left open by Bade, Dales, and Lykova (Mem. Amer. Math. Soc. 1999), while the second answers a question of Helemskii. The Banach space $E$ that we use was originally introduced by Read (J. London Math. Soc. 1989).