Certain Properties of n-Characters and n-Homomorphisms on Topological Algebras

TL;DR: In this article, the authors extend the notion of homomorphisms and characters to characters on algebras, and show that some properties of characters are also valid for continuous characters on commutative topological algebraic topologies.

Abstract: We extend the notion of homomorphisms and characters to $$n$$ -homomorphisms and $$n$$ -characters on algebras, and then show that some properties of characters are also valid for $$n$$ -characters on commutative $$lmc$$ topological algebras, and the space of continuous $$n$$ -characters $$M_{(A,n)}$$ is relatively compact in $$A'$$ (the dual space of $$A$$ ), with the weak* topology (Gelfand topology), whenever $$A$$ is a commutative $$lmc$$ $$Q$$ -algebra. We also find relations between characters, $$n$$ -characters, and continuous $$n$$ -characters on commutative Frechet algebras. Let $$B$$ be a topological algebra and $$(A_{\alpha },\varphi _{\beta \alpha })$$ (resp. $$(A_{\alpha },\varphi _{\alpha \beta })$$ ) be an inductive system (resp. a projective system) of topological algebras. Then we obtain relations between $$n-Hom(A_{\alpha },B)$$ and $$n-Hom(A,B)$$ , or between , the inductive limit, and $$M_{(A,n)}$$ , where , is the inductive limit (resp. $$A=\varprojlim A_{\alpha }$$ , is the projective limit) and $$n-Hom(A_{\alpha },B)$$ (resp. $$n-Hom(A,B)$$ ), is the space of all continuous n-homomorphisms from $$A_\alpha $$ (resp. $$A$$ ) into $$B$$ .