Lie Modules of Banach Space Nest Algebras

Pedro Capitão and Lina Oliveira ()
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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

Mathematics, 2024, vol. 12, issue 8, 1-16

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.

Keywords: Lie module; bimodule; nest algebra; Banach space (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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