The stability of the exponential equation

We generalize the well-known Baker's superstability result for exponential mappings with values in the field of complex numbers to the case of an arbitrary commutative complex semisimple Banach algebra. It was shown by Ger that the superstability phenomenon disappears if we formulate the stability question for exponential complex-valued functions in a more natural way. We improve his result by showing that the maximal possible distance of an $\varepsilon$-approximately exponential function to the set of all exponential functions tends to zero as $\varepsilon$ tends to zero. In order to get this result we have to prove a stability theorem for real-valued functions additive modulo the set of all integers $\mathbb{Z}$.