ON THE AUTOMORPHISMS OF THE SPECTRAL UNIT BALL.

Let A be a (complex, unital) semisimple Banach algebra and denote by Ω_A its open spectral unit ball, that is, the set Ω_A = {a ∈, A : σ(a) ⊆ D}, where σ (a) denotes the spectrum of a in A and D is the open unit disc in the complex plane. We prove that if F : Ω_A → Ω_A is a holomorphic map satisfying F (0) = 0 and F′ (0) = I (the identity of A), then for a in Ω_A the intersection of all closed discs lying inside D and containing σ (a) equals the intersection of all closed discs lying inside D and containing σ (F (a)). When all the elements of A have an at most countable spectrum and F is biholomorphic, this implies that F preserves the convex hull of the spectrum. As an application of the same equality, we prove that if B is a semisimple Banach algebra and T : A → B is a unital surjective spectral isometry, then σ (T (a)) = σ (a) in the case when σ (a) has exactly two elements. [ABSTRACT FROM AUTHOR]