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...

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Bibliographic Details
Published in:Journal of the London Mathematical Society 2013-08, Vol.88 (1), p.49-64
Main Authors: Gröchenig, Karlheinz, Klotz, Andreas
Format: Article
Language:English
Subjects:
Algebra
Estimates
Invariants
Inversions
Mathematical analysis
Norms
Smoothness
Spectra
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Summary: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.
ISSN:0024-6107
1469-7750
DOI:10.1112/jlms/jdt004