Approximately finite-dimensional Banach algebras are spectrally regular

Let B be a unital Banach algebra, which can in a certain sense be approximated by finite dimensional algebras. For instance, AF C⁎-algebras belong to this class. Further, let f be an analytic function on some bounded Cauchy domain Δ with values in B and suppose that the contour integral of the logar...

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Bibliographic Details
Published inLinear algebra and its applications Vol. 470; pp. 185 - 199
Main Authors Bart, Harm, Ehrhardt, Torsten, Silbermann, Bernd
Format Journal Article
LanguageEnglish
Published Elsevier Inc 01.04.2015
Subjects
AF [formula omitted]-algebra
AFD Banach algebra
Irrational rotation algebra
Logarithmic residue
secondary
AF C⁎-algebra
Irrational rotation algebra
AFD Banach algebra
Logarithmic residue
primary
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Summary:Let B be a unital Banach algebra, which can in a certain sense be approximated by finite dimensional algebras. For instance, AF C⁎-algebras belong to this class. Further, let f be an analytic function on some bounded Cauchy domain Δ with values in B and suppose that the contour integral of the logarithmic derivative f′(λ)f−1(λ) along the positively oriented boundary ∂Δ vanishes (or is even only quasinilpotent). We prove that then f takes invertible values on all of Δ. This means that such Banach algebras are spectrally regular.
ISSN:0024-3795
1873-1856
DOI:10.1016/j.laa.2014.06.023