Approximately finite-dimensional Banach algebras are spectrally regular
Dedicated to Leiba Rodman on the occasion of his 70th birthday, with admiration
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.
|Keywords||AF C∗-algebra, AFD Banach algebra, Irrational rotation algebra, Logarithmic residue|
|Persistent URL||dx.doi.org/10.1016/j.laa.2014.06.023, hdl.handle.net/1765/101577|
|Journal||Linear Algebra and Its Applications|
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