Norm-Controlled Inversion in Smooth Banach Algebras, I

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.