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WHEN IS A COMPLETION OF THE UNIVERSAL ENVELOPING ALGEBRA A BANACH PI-ALGEBRA?

Part of: Rings with polynomial identity Rings and algebras arising under various constructions Linear spaces and algebras of operators Special classes of linear operators

Published online by Cambridge University Press:  05 September 2022

O. YU. ARISTOV
O. YU. ARISTOV*
Affiliation:
Obninsk, Russia
*
e-mail: aristovoyu@inbox.ru
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Abstract

We prove that a Banach algebra B that is a completion of the universal enveloping algebra of a finite-dimensional complex Lie algebra $\mathfrak {g}$ satisfies a polynomial identity if and only if the nilpotent radical $\mathfrak {n}$ of $\mathfrak {g}$ is associatively nilpotent in B. Furthermore, this holds if and only if a certain polynomial growth condition is satisfied on $\mathfrak {n}$ .

Keywords

Banach PI-algebrauniversal enveloping algebranilpotent radicalpolynomial growthBanach space representation

MSC classification

Primary: 47L10: Algebras of operators on Banach spaces and other topological linear spaces
Secondary: 16R99: None of the above, but in this section 16S30: Universal enveloping algebras of Lie algebras 47B40: Spectral operators, decomposable operators, well-bounded operators, etc.
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 107 , Issue 3 , June 2023 , pp. 493 - 501
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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