On the Gleason-Kahane-Żelazko Theorem for Associative Algebras

Abstract

The classical Gleason-Kahane-Żelazko Theorem states that a linear functional on a complex Banach algebra not vanishing on units, and such that Lambda (textbf{1})=1, is multiplicative, that is, Lambda (ab)=Lambda (a)Lambda (b) for all a,bin A. We study the GKŻ property for associative unital algebras, especially for function algebras. In a GKŻ algebra A over a field of at least 3 elements, and having an ideal of codimension 1, every element is a finite sum of units. A real or complex algebra with just countably many maximal left (right) ideals, is a GKŻ algebra. If A is a commutative algebra, then the localization A_{P} is a GKŻ-algebra for every prime ideal P of A. Hence the GKŻ property is not a local-global property. The class of GKŻ algebras is closed under homomorphic images. If a function algebra Asubseteq {mathbb {F}}^{X} over a subfield {mathbb {F}} of {mathbb {C}}, contains all the bounded functions in {mathbb {F}}^{X}, then each element of A is a sum of two units. If A contains also a discrete function, then A is a GKŻ algebra. We prove that the algebra of periodic distributions, and the unitisation of the algebra of distributions with support in (0,infty ) satisfy the GKŻ property, while the algebra of compactly supported distributions does not.