Spectrally additive maps on Banach algebras

Abstract

Let A and B be complex unital Banach algebras, and let $\sigma(x)$ and $\hat{\sigma}(x)$ denote the spectrum of x and its polynomially convex hull, respectively. A spectrally additive map $\phi \colon A \to B$ is a surjective function which satisfies $\sigma(x + y) = \sigma(\phi(x) + \phi(y))$ for each $x, y \in A$. 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 $\phi \colon A \to B$ with the property that $\hat{\sigma}(x + y) = \hat{\sigma}(\phi(x) + \phi(y))$ for each $x, y \in A$ is a continuous Jordan-isomorphism.