Relation "commutator is a function" in Banach algebras

We discuss an analytic problem with roots in quantum group theory. When deforming commutation relations in universal enveloping algebras one sometimes encounters holomorphis functions that are not polynomials (for example, the $\mathrm{sinh}$ function). In the simplest case, the relation is of the form $xy-yx=h(y)$, where $h$ is a holomorphic function on some domain. If $h$ is not a polynomial, then it is natural to suppose that $x$ and $y$ are elements of a Banach algebra or an Arens-Michael algebra (i.e., a projective limit of Banach algebras). We show that the universal algebra generated by elements subject to the above condition is an analytic Ore extension. The construction involves a family $A_s$, $s\in(0,\infty]$, of local power series algebras. The appearence of $A_s$ follows from an asymptotic estimate for $\|(y-\lambda)^n\|$ as $n\to \infty$, where $\lambda$ is a zero of $h$, and $\|.\|$ is a submultiplicative seminorm (the estimate depends on the mutiplicity of $\lambda$). As an application, we consider the problem of embedding $A_s$ into holomorphically finitely generated algebras. In conclusion, some open problems will be discussed.