Topological groups of bounded homomorphisms on a topological group

Abstract

We consider a few types of bounded homomorphisms on a topological group. These classes of bounded homomorphisms are, in a sense, weaker than the class of continuous homomorphisms. We show that with appropriate topologies each class of these homomorphisms on a complete topological group forms a complete topological group.2010 Mathematics Subject Classification: 22A05, 54H11.

Решеточная Структура В Пространстве Ограниченных Гомоморфизмов Между Топологическими Решеточно Упорядоченными Кольцами

“…A homomorphism T : G → H is said to be positive if it maps positive elements of G into positive ones in H. Now, we recall some definition we need in the sequel (see [11] for further notifications about these facts). It should be mentioned here that in [11], the authors used the notion B(X, Y) for rings of all bounded group homomorphisms between topological rings; in this note, we replace it with Hom(X, Y) in compatible with [12] for homomorphisms as well as to show their nature as a homomorphism not an operator. Definition 1.Let X and Y be two topological rings.…”

“…In contrast with the case of all bounded homomorphisms between topological groups (considered in [12]), there are no more relations between these classes of bounded group homomorphisms between topological rings; see [11,…”

“…The set of all order bounded homomorphisms from G into H is denoted by Hom b (G, H). One may justify that under group operations of homomorphisms defined in[12] and invoking[5, Theorem 4.9], Hom b (G, H) is a group. Now, we prove a Riesz-Kantorovich formulae for order bounded homomorphisms compatible with[13, Theorem 1.18].…”