Norm-Controlled Inversion in Smooth Banach Algebras, I

TL;DR

This paper provides a quantitative analysis of spectral invariance in differential subalgebras of unital C*-algebras, establishing explicit norm estimates for invertible elements based on smoothness conditions.

Contribution

It introduces a quantitative version of spectral invariance, showing minimal smoothness guarantees norm control with explicit estimates.

Findings

Explicit differential norm estimates depending on condition number and norm ratios

Minimal smoothness via differential norms implies spectral invariance

Quantitative bounds enhance understanding of invertibility in Banach algebras

Abstract

Every differential subalgebra of a unital $C^*$-algebra is spectrally invariant. We derive a quantitative version of this well-known fact and show that a minimal amount of smoothness, as given by a differential norm, already implies norm control. We obtain an explicit estimate for the differential norm of an invertible element $a$. This estimate depends only on the condition number of $a$ and the ratio of two norms.