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Matematicheskie Zametki, 2021, Volume 109, Issue 3, Pages 323–337
DOI: https://doi.org/10.4213/mzm12746
(Mi mzm12746)
 

This article is cited in 2 scientific papers (total in 2 papers)

The Relation “Commutator Equals Function” in Banach Algebras

O. Yu. Aristov
Full-text PDF (570 kB) Citations (2)
References:
Abstract: The relation $xy-yx=h(y)$, where $h$ is a holomorphic function, occurs naturally in the definitions of some quantum groups. To attach a rigorous meaning to the right-hand side of this equality, we assume that $x$ and $y$ are elements of a Banach algebra (or of an Arens–Michael algebra). We prove that the universal algebra generated by a commutation relation of this kind can be represented explicitly as an analytic Ore extension. An analysis of the structure of the algebra shows that the set of holomorphic functions of $y$ degenerates, but at each zero of $h$, some local algebra of power series remains. Moreover, this local algebra depends only on the order of the zero. As an application, we prove a result about closed subalgebras of holomorphically finitely generated algebras.
Keywords: commutation relation, quantum group, Banach algebra, Arens–Michael algebra, analytic Ore extension, holomorphically finitely generated algebra.
Received: 06.04.2020
Revised: 06.10.2020
English version:
Mathematical Notes, 2021, Volume 109, Issue 3, Pages 323–334
DOI: https://doi.org/10.1134/S0001434621030019
Bibliographic databases:
Document Type: Article
UDC: 517.986
Language: Russian
Citation: O. Yu. Aristov, “The Relation “Commutator Equals Function” in Banach Algebras”, Mat. Zametki, 109:3 (2021), 323–337; Math. Notes, 109:3 (2021), 323–334
Citation in format AMSBIB
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  • https://www.mathnet.ru/eng/mzm/v109/i3/p323
  • This publication is cited in the following 2 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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