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.

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doi.org/10.1016/j.laa.2014.06.023, hdl.handle.net/1765/101577
Linear Algebra and Its Applications
Department of Econometrics

Bart, H., Ehrhardt, T., & Silbermann, B. (2015). Approximately finite-dimensional Banach algebras are spectrally regular. Linear Algebra and Its Applications, 470, 185–199. doi:10.1016/j.laa.2014.06.023