Weights on Discrete Semigroups and Topological Groups

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.