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A Note on Some Classes of Function Algebras

Murphy E. Egwe, Funke Yusuf

TL;DR

This note surveys a broad spectrum of function algebras, presenting foundational definitions and then detailing structural properties for each class. It covers Bergman, Bloch, and Hardy spaces alongside spherical-function algebras, Lie pseudogroups, and the Colombeau generalized-function framework, providing both new proofs and consolidated results. Key contributions include a Banach-algebra perspective on Bergman and Hardy spaces with operator-analytic tools (Toeplitz relations), a multiplier-algebra treatment of Bloch-type spaces, a topological-algebra view of spherical-function convolutions, and a rigorous Colombeau construction that connects classical function spaces with generalized functions via moderate/negligible nets. The work advances the theory by integrating classical complex-analytic algebras with modern operator-space, representation-theoretic, and Colombeau structures, enabling novel proofs and cross-domain techniques with potential applications in analysis and mathematical physics.

Abstract

In this note, we establish some new results on some special types of function algebras and also give new proofs to some existing ones

A Note on Some Classes of Function Algebras

TL;DR

This note surveys a broad spectrum of function algebras, presenting foundational definitions and then detailing structural properties for each class. It covers Bergman, Bloch, and Hardy spaces alongside spherical-function algebras, Lie pseudogroups, and the Colombeau generalized-function framework, providing both new proofs and consolidated results. Key contributions include a Banach-algebra perspective on Bergman and Hardy spaces with operator-analytic tools (Toeplitz relations), a multiplier-algebra treatment of Bloch-type spaces, a topological-algebra view of spherical-function convolutions, and a rigorous Colombeau construction that connects classical function spaces with generalized functions via moderate/negligible nets. The work advances the theory by integrating classical complex-analytic algebras with modern operator-space, representation-theoretic, and Colombeau structures, enabling novel proofs and cross-domain techniques with potential applications in analysis and mathematical physics.

Abstract

In this note, we establish some new results on some special types of function algebras and also give new proofs to some existing ones
Paper Structure (15 sections, 15 theorems, 140 equations)

This paper contains 15 sections, 15 theorems, 140 equations.

Table of Contents

  1. Introduction
  2. Some Classes of Operator Algebras
  3. Bi-invariant Function
  4. Holomorphic Functions
  5. The Space $\mathcal{L}^{p,\infty}$
  6. Smooth Manifold
  7. Algebra of Bergman Spaces
  8. Algebra of Bloch Spaces
  9. Algebra of Hardy Spaces
  10. Proof:
  11. Algebra of Spherical Functions.
  12. Algebra of Lie Pseudogroups
  13. Lie Algebra of Pseudogroups
  14. The Colombeau Algebra.
  15. $\mathcal{L}^{1}\left(\mathbb{R}^{n}\right)$ Embedded in the Colombeau Generalized Functions.

Key Result

Theorem 1.1

The algebra of bounded holomorphic functions with the sup norm is a Banach algebra.

Theorems & Definitions (38)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Theorem 1.1
  • Definition 1.7
  • Definition 2.1
  • Proposition 2.1
  • ...and 28 more