Let $B$ be a commutative, Jacobson semi-simple unital Banach algebra and take an invertible element $x$ in $B$. We may then compute the infimum of the Gelfand transform:


$\delta = \inf |f(x)|$


where the infimum is taken over all charaters on $B$. Can we estimate $\|x^{-1}\|$ from above by a function of $\|x\|$ and $\delta$?


This look very elementary but a solution escapes me.



Bounding the norm of the inverse in a commutative Banach algebra from above
This does not always work, although for some algebras where it does not work, one has a weaker form of controlled inversion where one fixes $\delta$ to be greater than some threshold, and can then bound $\Vert f^{-1}\Vert$ from above by some function depending only on $\delta$ (assuming $\inf\_x \vert f(x)\vert \geq\delta$). That is, provided we know $|f|$ is "bounded away from zero by a sufficiently large amount" that is enough to get an upper bound on the norm of $f^{-1}$.


Off the top of my head, I don't know the full history of the negative and positive results in this area, but a very good place to find out more is this long article of Nikolskii.


N. K. Nikolskii, *In search of the invisible spectrum.*
[Annales de l'Institut Fourier, Volume 49 (1999) no. 6, pp. 1925–1998](http://www.numdam.org/item/AIF_1999__49_6_1925_0/)


The title of Nikolskii's article comes from the Wiener-Pitt phenomenon: namely, there is a finite measure $\mu$ on ${\bf R}$ such that its Fourier transform is bounded uniformly away from zero as a function on $\widehat{{\bf R}}$, but $\mu$ is not invertible as an element of the Banach algebra $M({\bf R})$.


In a positive direction: the desired form of "controlled inversion" does hold for various commutative Banach algebras where the Gelfand transforms of elements have some kind of "differentiability" that is detected by the norm. Some interesting general results were obtained c. 2012 in work of Gröchenig and Klotz, see
<https://arxiv.org/abs/1207.1269> and <https://arxiv.org/abs/1211.2974> .


