The Gleason--Kahane--Zelazko theorem: a gentle introduction

After some personal recollections of Jean-Pierre Kahane, this paper gives a gentle introduction to the Gleason-Kahane-Zelazko theorem. The theorem states that a normalised complex-linear functional on a unital complex Banach algebra is multiplicative whenever it does not vanish on invertible elements. Neither continuity of the functional nor commutativity of the algebra is assumed. We give a detailed account of three proofs: the original argument of Gleason and, independently, Kahane and Zelazko; the polynomial proof of Roitman and Sternfeld; and a proof by the author. We explain the complex analysis behind each: a special case of Hadamard's factorisation theorem, a polynomial argument using the fundamental theorem of algebra, and the Pólya-Saxer theorem on zero-free entire functions and their derivatives. The elementary entire-function lemma needed for the first proof is proved in full; the Pólya-Saxer theorem is used without proof. We give full details of the algebraic argument common to the three proofs, which deduces multiplicativity from the preservation of squares, including the non-commutative case. We then discuss applications and counterexamples, and present an extension to Ransford's generalised spectra using the Roitman--Sternfeld argument. We conclude with a brief guide to the more recent literature.