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 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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