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.
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Notes
Here and in what follows, by “algebra” we mean “associative algebra with unit over the field \(\mathbb C\) of complex numbers.”
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Translated from Matematicheskie Zametki, 2021, Vol. 109, pp. 323-337 https://doi.org/10.4213/mzm12746.
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Aristov, O.Y. The Relation “Commutator Equals Function” in Banach Algebras. Math Notes 109, 323–334 (2021). https://doi.org/10.1134/S0001434621030019
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DOI: https://doi.org/10.1134/S0001434621030019