Hostname: page-component-669899f699-cf6xr Total loading time: 0 Render date: 2025-04-25T16:51:38.009Z Has data issue: false hasContentIssue false
  • English
  • Français

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 Special classes of linear operators Linear spaces and algebras of operators

Published online by Cambridge University Press:  05 September 2022

O. YU. ARISTOV
O. YU. ARISTOV*
Affiliation:
Obninsk, Russia
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Aljadeff, E., Giambruno, A., Procesi, C. and Regev, A., Rings with Polynomial Identities and Finite Dimensional Representations of Algebras, Colloquium Publications, 66 (American Mathematical Society,Providence, RI, 2020).Google Scholar
Aristov, O. Yu., ‘Holomorphic functions of exponential type on connected complex Lie groups’, J. Lie Theory 29(4) (2019), 1045–1070.Google Scholar
Aristov, O. Yu., ‘Functions of class ${C}^{\infty }$ in non-commuting variables in the context of triangular Lie algebras’, Izv. RAN Ser. Mat. 86(6) (2022), to appear (in Russian); English transl., Izv. Math. 86 (2022), to appear.Google Scholar
Aristov, O. Yu., ‘Length functions exponentially distorted on subgroups of complex Lie groups’, Preprint, 2022.Google Scholar
Aristov, O. Yu., ‘Decomposition of the algebra of analytic functionals on a connected complex Lie group and its completions into iterated analytic smash products’, Preprint, 2022 (in Russian).Google Scholar
Bahturin, Y. A., ‘The structure of a PI-envelope of a finite-dimensional Lie algebra’, Izv. Vyssh. Uchebn. Zaved. Mat. 1985(11) (1985), 60–62; English transl., Soviet Math. (Iz. VUZ) 29(11) (1985), 83–87.Google Scholar
Beltiţă, D. and Şabac, M., Lie Algebras of Bounded Operators, Operator Theory: Advances and Applications, 120 (Birkhäuser,Basel, 2001).Google Scholar
Bourbaki, N., Elements of Mathematics. Lie Groups and Lie Algebras. Part I: Chapters 1–3 (Addison-Wesley/Hermann,Paris, 1975).Google Scholar
Dixmier, J., Enveloping Algebras (North-Holland,Amsterdam, 1977).Google Scholar
Dosi, A., ‘Fréchet sheaves and Taylor spectrum for supernilpotent Lie algebra of operators’, Mediterr. J. Math. 6 (2009), 181–201.CrossRefGoogle Scholar
Dosi, A. A., ‘Taylor functional calculus for supernilpotent Lie algebra of operators’, J. Operator Theory 63(1) (2010), 191–216.Google Scholar
Dosi, A. A., ‘The Taylor spectrum and transversality for a Heisenberg algebra of operators’, Mat. Sb. 201(3) (2010), 39–62; English transl., Sb. Math. 201(3) (2010), 355–375.Google Scholar
Dosiev (Dosi), A. A., ‘Cohomology of sheaves of Fréchet algebras and spectral theory’, Funktsional. Anal. i Prilozhen. 39(3) (2005), 76–80; English transl., Funct. Anal. Appl. 39(3) (2005), 225–228.Google Scholar
Dosiev (Dosi), A. A., ‘Formally-radical functions in elements of a nilpotent Lie algebra and noncommutative localizations’, Algebra Colloq. 17(Spec. Iss. 1) (2010), 749–788.Google Scholar
Helemskii, A. Y., Banach and Polynormed Algebras: General Theory, Representations, Homology (Nauka,Moscow, 1989) (in Russian); English transl. (Oxford University Press, Oxford, 1993).Google Scholar
Herstein, I. N., Noncommutative Rings, Carus Mathematical Monographs, 15 (Mathematical Association of America/Wiley,New York, 1968).Google Scholar
Herstein, I. N., Small, L. and Winter, D. J., ‘A Lie algebra variation on a theorem of Wedderburn’, J. Algebra 144(2) (1991), 496–509.CrossRefGoogle Scholar
Kanel-Belov, A., Karasik, Y. and Rowen, L. H., Computational Aspects of Polynomial Identities, Volume I, Kemer’s Theorems, 2nd edn (Chapman and Hall/CRC,Boca Raton, FL, 2016).Google Scholar
Krupnik, N. Y., Banach Algebras with Symbol and Singular Integral Operators (Birkhäuser Verlag,Basel–Boston, 1987).CrossRefGoogle Scholar
Laursen, K. B. and Neumann, M. M., An Introduction to Local Spectral Theory, London Mathematical Society Monographs, 20 (Clarendon Press,Oxford, 2000).Google Scholar
Müller, V., ‘Nil, nilpotent and PI-algebras’, in: Functional Analysis and Operator Theory, Banach Center Publications, 30 (ed. Zemánek, J.) (PWN,Warsaw, 1994), 259–265.Google Scholar
Pirkovskii, A. Y., ‘Arens–Michael envelopes, homological epimorphisms, and relatively quasi-free algebras’, Tr. Mosk. Mat. Obs. 69 (2008), 34–125 (in Russian); English transl., Trans. Moscow Math. Soc. 2008 (2008), 27–104.Google Scholar
Taylor, J. L., ‘A general framework for a multi-operator functional calculus’, Adv. Math. 9 (1972), 183–252.CrossRefGoogle Scholar
Turovskii, Y. V., ‘On commutativity modulo the Jacobson radical of the associative envelope of a Lie algebra’, in: Spectral Theory of Operators and its Applications, Vol. 8 (ed. Maksudov, F. G.) (ELM,Baku, 1987), 199–211 (in Russian).Google Scholar