Banach Function Algebras, Arens Regularity, and BSE Norms

✍ Scribed by Harold Garth Dales, Ali Ülger

• Publisher: Springer
• Year: 2024
• Tongue: English
• Leaves: 452
• Series: CMS/CAIMS Books in Mathematics; 12
• Category: Library

✦ Synopsis

This book is about semisimple Banach algebras with a focus on those that are commutative.

After laying out the necessary background material from functional analysis, geometry of Banach spaces and measure theory, we introduce many specific Banach algebras from operator theory, harmonic analysis, and function theory and study their basic properties.

Some of the questions dealt with in the book are: Whether the introduced Banach algebras are BSE-algebras, whether they have BSE norms, whether they have the separating ball property or some variant of it, and whether they are Arens regular. The book contains quite a few new results, as well as new proofs of a good many known results.

The book is intended for those who are preparing to work in Banach algebras or who have been doing research in related areas.

✦ Table of Contents

Preface
Acknowledgements
Contents
1 Banach spaces and operators
1.1 Basic notation
1.2 Banach spaces
1.3 Bounded linear operators
1.4 Tensor products
2 Banach algebras
2.1 Algebras and Banach algebras
2.2 C*-algebras and von Neumann algebras
2.3 Biduals of Banach algebras
2.4 Dual Banach algebras
3 Banach function algebras
3.1 Banach function algebras
3.2 Banach sequence algebras
3.3 Projective tensor products of Banach function algebras
3.4 The separating ball property
3.5 Pointwise approximate identities
3.6 Uniform algebras and Gleason parts
4 Banach algebras on locally compact groups
4.1 Group and measure algebras
4.2 Locally compact abelian groups
4.3 Fourier and Fourier–Stieltjes algebras
4.4 Figà-Talamanca–Herz algebras
5 BSE norms and BSE algebras
5.1 The space L(A) and the algebra mathcalQ(A)
5.2 Basic definitions and results about BSE norms
5.3 Examples
5.4 Pointwise contractive Banach function algebras and ell1-norms
5.5 Banach function algebras that are ideals in their bidual
6 Arens regularity
6.1 Topological centres
6.2 Arens regularity of projective tensor products
6.3 Biduals of semigroup algebras
6.4 Arens regularity for algebras on locally compact groups
References
Index of Terms
Index of Theorems and Examples
Index of Symbols