Spectrally additive maps on Banach algebras

Let A and B be complex unital Banach algebras, and let σ(x) and σ^ (x) denote the spectrum of x and its polynomially convex hull, respectively. A spectrally additive map ϕ: A→ B is a surjective function which satisfies σ(x+ y) = σ(ϕ(x) + ϕ(y)) for each x, y∈ A . By establishing a new additive characterization of rank one elements in a Banach algebra, we prove that any spectrally additive map acting on a semisimple domain preserves rank one elements in both directions. This settles an open question raised in [1], which ultimately then classifies spectrally additive maps on a large class of Banach algebras as Jordan-isomorphisms. By refining the techniques in [1] even further, we are able to prove the following more general result: If A is semisimple and either A or B has an essential socle, then any surjective map ϕ: A→ B with the property that σ^ (x+ y) = σ^ (ϕ(x) + ϕ(y)) for each x, y∈ A is a continuous Jordan-isomorphism.