Weights on Discrete Semigroups and Topological Groups

Thèse de Master , 2008 , 63 Pages

Mathématiques - Analyse

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

• Abstract
• Acknowledgements
• 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
• Bibliography

Objective & Thematic Focus

This dissertation aims to systematically develop methods for constructing "weights" on discrete semigroups and topological groups. The research explores the nature of these weights, which are strictly positive, submultiplicative functions, and their crucial role in forming Banach algebras, specifically weighted discrete semigroup algebras l¹(S, w).

• Construction of weights on discrete semigroups.
• Construction of measurable weights on topological groups.
• Analysis of submultiplicative functions ("weights") in algebraic structures.
• Exploration of generalized semicharacters and their relation to weights.
• Investigation into the influence of weights on the structure of Banach algebras.
• Methods involving subadditive maps and combining existing weights.

Excerpt from the Book

Weights on Semigroups

Definition 1.3.1. Let S be a semigroup. A weight on a semigroup S is a positive map w : S → (0, ∞) such that w(st) ≤ w(s)w(t) (s, t ∈ S).

Definition 1.3.2. A weight w on a semigroup S is: (1) radical if lim_n→∞w(sⁿ)^1/n = 0 (s ∈ S); (2) semisimple if lim_n→∞w(sⁿ)^1/n ≠ 0 (s ∈ S).

Lemma 1.3.3. Let w₁ and w₂ be weights on a semigroup S. For each s ∈ S, define w(s) = w₁(s)w₂(s). Then

1. w is a weight on S;
2. if w₁ or w₂ is radical, then w is radical;
3. w is semisimple iff both w₁ and w₂ are semisimple.

Proof: 1 This is clear. 2 and 3 will follow from the following identity. For s ∈ S, we have lim_n→∞w(sⁿ)^1/n = lim_n→∞w₁(sⁿ)^1/n · w₂(sⁿ)^1/n = lim_n→∞w₁(sⁿ)^1/n · lim_n→∞w₂(sⁿ)^1/n.

Remark 1.3.4. Let λ be a positive irrational number. Let S = Z⁺×Z⁺ \ {0}. Define w₁(m + λn) = {e^-m² if m≠ 0; 1 if m = 0.} and w₂(m + λn) = {e^-n² if n≠0; 1 if n = 0.} Then w₁(kλ)^1/k = 1^1/k = 1 and w₂(k)^1/k = 1/k for all k ∈ N. Hence neither w₁ nor w₂ is radical. However, for any k ∈ N and m + λn ∈ S, w(k(m + λn)) = w(km + λkn) ≤ w(km)·ω(λkn) = w₁(km) · ω₂(λkn) lim_k→∞ w(k(m + λn))^1/k = lim_k→∞ w₁(km)^1/k · lim_k→∞ w₂(λkm)^1/k. Since m + λn ≠ 0, either m ≠ 0 or n ≠ 0. Hence, either lim_k→∞w₁(km)^1/k = 0 or lim_k→∞w₁(km)^1/k = 0. So w is radical. Thus, the converse of Lemma 1.3.3(2) is not true.

Summary of Chapters

Chapter 1: Introduction: This chapter introduces fundamental definitions from semigroup theory, including various types of semigroups, generalized semicharacters, and the core concept of a "weight" on a semigroup.

Chapter 2: Weights on Discrete Semigroups: This chapter focuses on developing methods to construct weights on discrete semigroups, utilizing subadditive maps and exploring how new weights can be derived from existing ones, with examples on specific semigroups.

Chapter 3: Weights on Topological Groups: This chapter extends the study to topological groups, constructing measurable weights on these groups, including classical linear groups and compactly generated groups, relevant for Burling algebras.

Keywords

Semigroups, Weights, Topological Groups, Discrete Semigroups, Subadditive Maps, Generalized Semicharacters, Banach Algebras, Submultiplicativity, Radical Weights, Semisimple Weights, Measurable Weights, Locally Compact Groups, Polynomial Growth.

Frequently Asked Questions

What is the main topic of this work?

This work primarily focuses on the construction and properties of "weights" on discrete semigroups and topological groups, and their application in the theory of Banach algebras.

What are the central thematic areas covered?

The central thematic areas include semigroup theory, topological group theory, the definition and properties of weights (submultiplicative functions), generalized semicharacters, and the construction of Banach algebras like weighted discrete semigroup algebras.

What is the primary objective or research question?

The primary objective is to develop various methods for constructing weights on both discrete semigroups and topological groups, investigating how these weights influence algebraic structures.

Which scientific method is employed?

The work employs a theoretical and deductive mathematical approach, focusing on definitions, lemmas, theorems, and proofs to construct and characterize different types of weights.

What is discussed in the main body?

The main body delves into the definitions of semigroups, generalized semicharacters, and weights. It then explores various methods for constructing weights on discrete semigroups using (sub)additive maps and deriving new weights from existing ones. Finally, it extends these concepts to topological groups, including measurable weights and classical linear groups.

What key terms characterize this work?

Key terms characterizing this work are: Semigroups, Weights, Topological Groups, Subadditive Maps, Generalized Semicharacters, Banach Algebras, Submultiplicativity, Radical Weights, Semisimple Weights, Measurable Weights, Locally Compact Groups.

How is a "weight" formally defined in the context of this dissertation?

A weight on a semigroup S is formally defined as a strictly positive function w: S → (0, ∞) that satisfies the submultiplicative inequality: w(st) ≤ w(s)w(t) for all s, t ∈ S.

What is the significance of "submultiplicativity" for a weight function?

Submultiplicativity is the defining property of a weight function, ensuring that the "size" of a product of elements is bounded by the product of their individual "sizes," which is crucial for constructing Banach algebras like l¹(S, w).

How are weights constructed on discrete semigroups using (sub)additive maps?

Weights are constructed by defining a function w(s) = e^λη(s) where η: S → ℝ is a subadditive map and λ > 0. The subadditivity of η directly translates to the submultiplicativity of w, thus forming a weight.

What types of semicharacters are introduced, and how do they relate to weights?

Generalized semicharacters, positive generalized semicharacters, bounded semicharacters, and w-bounded semicharacters are introduced. A w-bounded semicharacter θ on S satisfies |θ(s)| ≤ w(s) for all s ∈ S, linking semicharacters to the bounding property of weights.

• Université: Sardar Patel University (Gujarat Arts and Science College, Ahmedabad)
• Auteur: Bhavin Mansukhlal Patel (Auteur)
• Année de publication: 2008
• Pages: 63
• N° de catalogue: V1665981
• ISBN (PDF): 9783389161630
• ISBN (Livre): 9783389161647
• Langue: anglais
• mots-clé: Weights, Semigroup, Topological group