Geodesic
The Relation ``Commutator Equals Function'' in Banach Algebras
Matematičeskie zametki, Tome 109 (2021) no. 3, pp. 323-337

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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.
Mots-clés : commutation relation
Keywords: quantum group, Banach algebra, Arens–Michael algebra, analytic Ore extension, holomorphically finitely generated algebra. 
@article{MZM_2021_109_3_a0,
     author = {O. Yu. Aristov},
     title = {The {Relation} {``Commutator} {Equals} {Function''} in {Banach} {Algebras}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {323--337},
     publisher = {mathdoc},
     volume = {109},
     number = {3},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2021_109_3_a0/}
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O. Yu. Aristov. The Relation ``Commutator Equals Function'' in Banach Algebras. Matematičeskie zametki, Tome 109 (2021) no. 3, pp. 323-337. http://geodesic.mathdoc.fr/item/MZM_2021_109_3_a0/