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 (Formula presented.) be a Banach space nest algebra and (Formula presented.) be a weakly closed Lie (Formula presented.) -module. We show that there exist a weakly closed (Formula presented.) -bimodule (Formula presented.), a weakly closed subalgebra (Formula presented.) of (Formula presented.), and a largest weakly closed (Formula presented.) -bimodule (Formula presented.) (Formula presented.), (Formula presented.). The first inclusion holds in general, whilst the second is shown to be valid in a class of nest algebras.

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Centrum Wiskunde & Informatica, Amsterdam (CWI), The Netherlands

Capitao, P., & Oliveira, L. (2024). Lie modules of Banach space nest algebras. Mathematics, 12(8), 1251:1–1251:16. doi:10.3390/math12081251