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

We show that there exists a Banach space E such that: the Banach algebra B(E) of bounded, linear operators on E has a singular extension that splits algebraically, but it does not split strongly; the homological bidimension of B(E) is at least two. The first of these conclusions solves a natural problem left open by Bade, Dales, and Lykova (1999) [1], while the second answers a question of Helemskii. The Banach space E that we use was originally introduced by Read (1989) [9]. (C) 2016 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.