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Lie Modules of Banach Space Nest Algebras

Author

Listed:
  • Pedro Capitão

    (Centrum Wiskunde & Informatica, Science Park 123, 1098XG Amsterdam, The Netherlands)

  • Lina Oliveira

    (Center for Mathematical Analysis, Geometry and Dynamical Systems, Department of Mathematics, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal)

Abstract

In the present work, we extend to Lie modules of Banach space nest algebras a well-known characterisation of Lie ideals of (Hilbert space) nest algebras. Let A be a Banach space nest algebra and L be a weakly closed Lie A -module. We show that there exist a weakly closed A -bimodule K , a weakly closed subalgebra D K of A , and a largest weakly closed A -bimodule J contained in L , such that J ⊆ L ⊆ K + D K , with [ K , A ] ⊆ L . The first inclusion holds in general, whilst the second is shown to be valid in a class of nest algebras.

Suggested Citation

  • Pedro Capitão & Lina Oliveira, 2024. "Lie Modules of Banach Space Nest Algebras," Mathematics, MDPI, vol. 12(8), pages 1-16, April.
  • Handle: RePEc:gam:jmathe:v:12:y:2024:i:8:p:1251-:d:1379538
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