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).Nous démontrons qu'il existe un espace de Banach tel que:• l'algèbre de Banach B(E) des opérateurs linéaires bornés sur E a une extension singulière qui scinde algébriquement mais qui ne scinde pas fortement;• la bidimension homologique de B(E) est au moins deux. La première de ces conclusions complète les résultats de Bade, Dales et Lykova (Mem. Amer. Math. Soc. 1999), et la seconde répond à une question de Helemskii. L'espace de Banach E a été introduit initialement par Read (J. London Math. Soc. 1989).

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Read produced the first example of a Banach space E R such that the associated Banach algebra B(E R ) of bounded operators admits a discontinuous derivation (J. London Math. Soc. 1989). We generalize Read's main theorem about B(E 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 sequencedenotes the ideal of weakly compact operators on E R , while ℓ ∼ 2 is the unitization of the Hilbert space ℓ 2 , endowed with the zero product.

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