The relation "commutator equals function'' in Banach algebras

TL;DR

This paper studies the algebraic structure of elements in Banach algebras satisfying a commutation relation involving a holomorphic function, revealing explicit representations and local algebraic behaviors at zeros of the function.

Contribution

It provides an explicit representation of the universal algebra generated by the commutation relation as an analytic Ore extension and analyzes local algebraic structures at zeros of the holomorphic function.

Findings

Universal algebra can be represented as an analytic Ore extension.

The set of holomorphic functions degenerates except at zeros of h.

Local algebra depends only on the order of the zero.

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.