Comptes Rendus
no. 5
Mathematical analysis/Functional analysis
Splittings of extensions and homological bidimension of the algebra of bounded operators on a Banach space
[Scissions d'extensions et bidimension homologique de l'algèbre d'opérateurs bornés sur un espace de Banach]

Présenté par : Gilles Pisier

Niels Jakob Laustsen1 ; Richard Skillicorn1
1 Department of Mathematics and Statistics, Fylde College, Lancaster University, Lancaster LA1 4YF, United Kingdom
Comptes Rendus. Mathématique, Volume 354 (2016) no. 5, pp. 459-463.

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 (1999) [1], tandis que la seconde répond à une question de Helemskii. L'espace de Banach E a été introduit initialement par Read (1989) [9].

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].

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2015.12.020
Niels Jakob Laustsen 1 ; Richard Skillicorn 1

1 Department of Mathematics and Statistics, Fylde College, Lancaster University, Lancaster LA1 4YF, United Kingdom 
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Niels Jakob Laustsen; Richard Skillicorn. Splittings of extensions and homological bidimension of the algebra of bounded operators on a Banach space. Comptes Rendus. Mathématique, Volume 354 (2016) no. 5, pp. 459-463. doi : 10.1016/j.crma.2015.12.020. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.12.020/

[1] W.G. Bade; H.G. Dales; Z.A. Lykova Algebraic and strong splittings of extensions of Banach algebras, Mem. Amer. Math. Soc., Volume 137 (1999)

[2] H.G. Dales; R.J. Loy; G.A. Willis Homomorphisms and derivations from B(E), J. Funct. Anal., Volume 120 (1994), pp. 201-219

[3] A.Ya. Helemskii The Homology of Banach and Topological Algebras, Kluwer, Dordrecht, The Netherlands, 1989

[4] A.Ya. Helemskii 31 problems of the homology of the algebras of analysis, Linear Complex Analysis Problem Book III, Part I, Lect. Notes Math., vol. 1573, Springer-Verlag, 1994, pp. 54-78

[5] B.E. Johnson Continuity of homomorphisms of algebras of operators, J. Lond. Math. Soc. (2), Volume 42 (1967), pp. 537-541

[6] B.E. Johnson The Wedderburn decomposition of Banach algebras with finite dimensional radical, Amer. J. Math., Volume 90 (1968), pp. 866-876

[7] B.E. Johnson Symmetric amenability and the nonexistence of Lie and Jordan derivations, Math. Proc. Camb. Philos. Soc., Volume 120 (1996), pp. 455-473

[8] N.J. Laustsen; R. Skillicorn Extensions and the weak Calkin algebra of Read's Banach space admitting discontinuous derivations (submitted) | arXiv

[9] C.J. Read Discontinuous derivations on the algebra of bounded operators on a Banach space, J. Lond. Math. Soc. (2), Volume 40 (1989), pp. 305-326

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