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On the primitive ideals of nest algebras

Part of: Linear spaces and algebras of operators Radicals and radical properties of rings

Published online by Cambridge University Press:  21 July 2020

John Lindsay Orr
John Lindsay Orr*
Affiliation:
Toll House, Traquair Road, InnerleithenEH44 6PF, UK (me@johnorr.us)
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Abstract

We show that Ringrose's diagonal ideals are primitive ideals in a nest algebra (subject to the continuum hypothesis). This answers an old question of Lance and provides for the first time concrete descriptions of enough primitive ideals to obtain the Jacobson radical as their intersection. Separately, we provide a standard form for all left ideals of a nest algebra, which leads to insights into the maximal left ideals. In the case of atomic nest algebras, we show how primitive ideals can be categorized by their behaviour on the diagonal and provide concrete examples of all types.

Keywords

nest algebraprimitive idealsnetscontinuum hypothesis

MSC classification

Primary: 47L35: Nest algebras, CSL algebras
Secondary: 16N20: Jacobson radical, quasimultiplication
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 63 , Issue 3 , August 2020 , pp. 737 - 760
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Copyright
Copyright © The Authors, 2020. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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