Gelfand's Theorem Unplugged

摘要

It was Norbert Wiener who made the observation that if a non-vanishing continuous function on the circle has an absolutely convergent Fourier series, then so does its reciprocal. Israel Gelfand saw in this picture the concept of a commutative Banach algebra, and extended Wiener's observation to these Banach algebra elements, with a completely different proof based on "maximal ideals". In this note we claim that the several variable spectral mapping theorem is able to unplug the Gelfand theory from the life support system of these elusive maximal ideals.