Kowalski-Slodkowski theorem for spectrum variants

Let A be a complex unital commutative Banach algebra. Let ϕ : A → C be a map such that for x, y ∈ A, ϕ(x) − ϕ(y) ∈ σε (x − y) and ϕ has C−linear differential almost everywhere. Then ϕ is approximately multiplicative. A similar conclusion was reached by replacing the differential condition with comparable assumptions on the map. This result is similar to the Kowalski-Slodkowski theorem. Analogous versions of it were discussed for the exponential spectrum and a particular class of the Ransford spectrum.