On the Gleason-Kahane-\.{Z}elazko theorem for associative algebras

Abstract

The classical Gleason-Kahane-\.{Z}elazko Theorem states that a linear functional on a complex Banach algebra not vanishing on units, and such that $\Lambda(\mathbf 1)=1$, is multiplicative, that is, $\Lambda(ab)=\Lambda(a)\Lambda(b)$ for all $a,b\in A$. We study the GK\.Z property for associative unital algebras, especially for function algebras. In a GK\.Z 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 GK\.Z algebra. If $A$ is a commutative algebra, then the localisation $A_{P}$ is a GK\.Z-algebra for every prime ideal $P$ of $A$. Hence the GK\.Z property is not a local-global property. The class of GK\.Z algebras is closed under homomorphic images. If a function algebra $A\subseteq \mathbb F^{X}$ 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 GK\.Z algebra. We prove that the algebra of periodic distributions, and the unitisation of the algebra of distributions with support in $(0,\infty)$ satisfy the GK\.Z property, while the algebra of compactly supported distributions does not.