Kowalski–Słodkowski Theorem for Spectrum Variants

Abstract

Let \(\mathcal A\) be a complex unital commutative Banach algebra. Let \(\varphi\colon \mathcal A\to \mathbb C\) be a map such that for \(x,y\in\mathcal A\), \(\varphi(x)-\varphi(y)\in\sigma_\varepsilon(x-y)\) and \(\varphi\) has \(\mathbb C\)-linear differentials almost everywhere. Then \(\varphi\) is approximately multiplicative. A similar conclusion is reached by replacing the differential condition with comparable assumptions on the map. This result is similar to the Kowalski–Słodkowski theorem. Analogous versions of it are also discussed for the exponential spectrum and for a particular class of the Ransford spectrum.