Approximately finite-dimensional Banach algebras are spectrally regular

Let BB be a unital Banach algebra, which can in a certain sense be approximated by finite dimensional algebras. For instance, AF C⁎C⁎-algebras belong to this class. Further, let f   be an analytic function on some bounded Cauchy domain Δ with values in BB and suppose that the contour integral of the logarithmic derivative f′(λ)f−1(λ)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.