On the Gleason-Kahane-Żelazko Theorem for Associative Algebras

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