There are various types of weights scattered in the mathematics literature. Here we attempt to develop some methods to construct weights on discrete semigroups. Then we also develop methods to construct weights on topological groups.
Consider a strictly positive function ω on a semigroup (S, ∗) satisfying the following simple inequality (so-called, submultiplicativity): (s ∗ t) ≤ ω(s)ω(t) (s, t ∈ S).
Such function ω is called a weight on S. The pair (S, ω), so-called a semigroup with weight, plays a very important roll in constructing a class of Banach algebras; namely the weighted discrete semigroup algebra 1(S, ω). The Banach algebra structure of the algebra 1(S, ω) is influenced by these two simple objects S and ω. So it is important to develop methods of constructing weights on semigroups. There are various types of weights scattered in the mathematics literature. Here we attempt to develop some methods to construct weights on discrete semigroups. Then we also develop methods to construct weights on topological groups.
Table of Contents
1 Introduction
1.1 Semigroups
1.2 Generalized Semicharacters
1.3 Weights on Semigroups
2 Weights on Discrete Semigroups
2.1 Weights using (Sub)additive Maps
2.2 New Weights from the Old Weights
2.3 Weights on Specific Semigroups
2.4 Other Methods of Constructing Weights
3 Weights on Topological groups
3.1 Topological Groups
3.2 Measurable Weights on Topological Groups
3.3 Weights on Classical Linear Groups
3.4 Weights on Compactly Generated Groups
Research Objectives and Core Topics
The primary objective of this dissertation is to systematically develop various methodologies for constructing weights on discrete semigroups and topological groups. By analyzing the properties of submultiplicative functions, the work explores how these weights influence the Banach algebra structure of weighted semigroup algebras.
- Mathematical construction of weights using subadditive and additive maps.
- Exploration of weight properties on discrete semigroups and specific algebraic structures.
- Development of measurable weights on locally compact topological groups.
- Analysis of weighted group algebras and the role of generalized characters.
- Classification of weights on compactly generated groups with polynomial growth.
Excerpt from the Book
Weights using (Sub)additive Maps
In this section, we shall construct weights using subadditive and additive maps on a semigroup S. In order to this, we shall make sure that there are many subadditive maps on S. The next result is about this.
Theorem 2.1.1. Let S be a semigroup.
1. Let η : S → [0, ∞) be a constant function. Then η is a subadditive map on S
2. Let S = for some non-empty set U ⊆ S. Set ηU (s) := inf{n ∈ N : s ∈ Un} (s ∈ S). Then ηU is a subadditive map on S.
3. Let η : S → [2, ∞) be a subadditive map. Then log η is a subadditive map on S.
4. Let η : S → [0, ∞) be a subadditive map and let 0 < p ≤ 1. Set ηp(s) = η(s)p (s ∈ S). Then ηp is a subadditive map on S.
5. Let S ba unital. Let d be an invariant metric on S. Set η(s) = d(s, 1) (s ∈ S). Then η is a subaditive map on S.
Chapter Summaries
1 Introduction: Provides foundational definitions from semigroup theory and introduces the core concept of a weight on a semigroup as the main topic of the dissertation.
2 Weights on Discrete Semigroups: Focuses on the construction of weights using various subadditive and additive maps, and investigates how new weights can be derived from existing ones.
3 Weights on Topological groups: Extends the study to topological groups, focusing on the construction of measurable weights and their application to Burling algebras on locally compact groups.
Keywords
Semigroups, Weights, Submultiplicativity, Banach algebras, Topological groups, Subadditive maps, Additive maps, Measurable weights, Locally compact groups, Generalized characters, Haar measure, Compactly generated groups, Polynomial growth, Semisimple weights, Weighted group algebras.
Frequently Asked Questions
What is the fundamental focus of this research?
The work focuses on the construction and properties of weights on discrete semigroups and topological groups, specifically investigating how these weights contribute to the structure of associated weighted Banach algebras.
What are the primary themes discussed?
Key themes include submultiplicative functions, measurable weights on locally compact groups, generalized characters, and the classification of weights on compactly generated groups with polynomial growth.
What is the core research objective?
The primary goal is to establish systematic methods for building weight functions on various algebraic structures, which are essential for the study of weighted discrete semigroup algebras and Burling algebras.
Which scientific methods are employed?
The research primarily utilizes analytical methods from the theory of semigroups, topological groups, and functional analysis, specifically applying subadditive and additive map constructions.
What is covered in the main body of the work?
The main body treats the derivation of weights via mapping properties, the extension of these concepts to topological and locally compact groups, and the examination of specific cases such as classical linear groups and groups with polynomial growth.
Which keywords best characterize this work?
The work is characterized by terms such as weights, semigroups, topological groups, submultiplicativity, measurable weights, and generalized characters.
What is the significance of the "submultiplicativity" condition?
Submultiplicativity is the defining property of a weight function on a semigroup (ω(st) ≤ ω(s)ω(t)), which ensures the proper Banach algebra structure of the resulting weighted algebras.
How are weights constructed for topological groups?
Weights on topological groups are constructed similarly to those on semigroups, with the additional requirement that they must also be measurable functions, leveraging the existing semigroup methods.
- Quote paper
- Bhavin Mansukhlal Patel (Author), 2008, Weights on Discrete Semigroups and Topological Groups, Munich, GRIN Verlag, https://www.grin.com/document/1665981
