On the Gleason-Kahane-Zelazko Theorem for Associative Algebras

The classical Gleason-Kahane-Zelazko Theorem states that a linear functional on a complex Banach algebra not vanishing on units, and such that Lambda(1) = 1, is multiplicative, that is, Lambda(ab) = Lambda(a)Lambda(b) for all a, b is an element of A. We study the GKZ property for associative unital algebras, especially for function algebras. In a GKZ 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 GKZ algebra. If A is a commutative algebra, then the localization A p is a GKZ-algebra for every prime ideal P of A. Hence the GKZ property is not a local-global property. The class of GKZ algebras is closed under homomorphic images. If a function algebra A subset of F-X over a subfield F of C, contains all the bounded functions in F-X, then each element of A is a sum of two units. If A contains also a discrete function, then A is a GKZ algebra. We prove that the algebra of periodic distributions, and the unitisation of the algebra of distributions with support in (0, infinity) satisfy the GKZ property, while the algebra of compactly supported distributions does not.