On the Gleason-Kahane-Żelazko Theorem for Associative Algebras

Abstract

AbstractThe classical Gleason-Kahane-Żelazko Theorem states that a linear functional on a complex Banach algebra not vanishing on units, and such that $$\Lambda (\textbf{1})=1$$ Λ ( 1 ) = 1 , is multiplicative, that is, $$\Lambda (ab)=\Lambda (a)\Lambda (b)$$ Λ ( a b ) = Λ ( a ) Λ ( b ) for all $$a,b\in A$$ a , b ∈ 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 $$A_{P}$$ 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 $$A\subseteq {\mathbb {F}}^{X}$$ A ⊆ F X over a subfield $${\mathbb {F}}$$ F of $${\mathbb {C}}$$ C , contains all the bounded functions in $${\mathbb {F}}^{X}$$ 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 $$(0,\infty )$$ ( 0 , ∞ ) satisfy the GKŻ property, while the algebra of compactly supported distributions does not.