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

TL;DR

This paper explores the Gleason-Kahane-Żelazko property in associative unital algebras, characterizing when elements are sums of units and examining the property in various algebra classes including function and distribution algebras.

Contribution

It extends the GKZ property to broader classes of associative algebras, providing new criteria and examples, especially for function and distribution algebras.

Findings

In GKZ algebras, every element can be expressed as a finite sum of units.

Certain function algebras contain all bounded functions and have elements as sums of two units.

Distribution algebras like periodic distributions satisfy the GKZ property, unlike compactly supported distributions.

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.