The quest to confirm Kaplansky’s conjecture from 1970 stating that every linear, unital, surjective, invertibility-preserving mapping between semisimple Banach algebras is a Jordan homomorphism has led to a plethora of beautiful results, notably by Aupetit and collaborators. The more general classes of spectrally bounded and spectrally isometric operators have also attracted a lot of attention. Many of the techniques involve methods from Complex Analysis and are surprisingly subtle. In this talk I will emphasise the usefulness of these methods and present some more recent results as well.
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