Isolated points of spaces of homomorphisms from ordered AL-algebras
Volume 577 / 2022
Abstract
We study isolated points of spaces comprised of homomorphisms between a given ordered AL-algebra and a given unital normed algebra. An ordered AL-algebra is a complex AL-space and simultaneously a Banach algebra such that the positive cone associated with the underlying partial ordering is closed under multiplication, and such that the algebra norm restricted to the positive cone is multiplicative. The class of ordered AL-algebras contains—as particular subclasses—semigroup algebras, group algebras, and convolution algebras of integrable, even functions on groups. We determine isolated points for various spaces of homomorphisms from ordered AL-algebras, including specifically spaces of homomorphisms from algebras belonging to the three subclasses just mentioned. We also discuss certain properties of homomorphisms beyond isolability, which one is naturally led to consider in connection with isolated points of spaces of homomorphisms. By way of application, we exhibit several AL-algebras that are not pairwise isometrically algebra and order isomorphic.
