Laustsen, Niels Jakob and Skillicorn, Richard (2016) Splittings of extensions and homological bidimension of the algebra of bounded operators on a Banach space. Comptes Rendus Mathématique, 354 (5). pp. 459-463. ISSN 1631-073X
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Abstract
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 which 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 (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).