[Paper Review] Frechet algebras in abstract Harmonic analysis

Zahra Alimohammadi, Ali Rejali|arXiv (Cornell University)|Nov 27, 2018

Holomorphic and Operator Theory43 references4 citations

TL;DR

This paper provides a comprehensive survey of Fréchet algebras in abstract harmonic analysis, comparing them to Banach algebras and generalizing key concepts such as amenability, Arens regularity, and Segal algebras. It establishes foundational results on the continuity of homomorphisms, the structure of multiplier algebras, and the equivalence of Arens regularity conditions in weighted algebras over groups and semigroups.

ABSTRACT

We provide a survey of the similarities and differences between Banach and Frechet algebras including some known results and examples. We also collect some important generalizations in abstract harmonic analysis; for example, the generalization of the concepts of vector-valued Lipschitz algebras, abstract Segal algebras, Arens regularity, amenability, weak amenability, ideal amenability, etc.

Motivation & Objective

• To compare and contrast Fréchet algebras with Banach algebras in the context of abstract harmonic analysis.
• To generalize classical concepts—such as amenability, weak amenability, ideal amenability, and biprojectivity—from Banach to Fréchet algebras.
• To investigate the continuity of homomorphisms between Fréchet algebras, particularly under semisimplicity and epimorphism conditions.
• To extend the theory of Segal algebras and multiplier algebras to the Fréchet setting, including topological properties and ideal structures.
• To explore the relationship between Arens regularity of a Fréchet algebra and its bidual, especially in weighted group and semigroup algebras.

Proposed method

• Utilizes the framework of locally convex topological algebras with a countable fundamental system of seminorms to define Fréchet algebras.
• Applies the theory of hypo-continuous multiplication in locally multiplicatively convex (lmc) algebras to establish that Fréchet algebras admit well-defined Arens products on their biduals.
• Employs the concept of bicommutative algebras to generalize results on the bidual and unit elements in the Arens product structure.
• Applies the notion of 0-clustered functions to characterize Arens regularity in weighted algebras over groups and semigroups.
• Uses the strict, uniform, and compact-open topologies on multiplier algebras to study Segal algebra structures in Fréchet algebras.
• Leverages results from Pirkovskii and others on flat cyclic modules and approximate contractibility to extend homological properties to Fréchet algebras.

Experimental results

Research questions

• RQ1 Under what conditions is a surjective homomorphism between Fréchet algebras continuous, especially when the codomain is semisimple?
• RQ2 To what extent can the classical theory of amenability, weak amenability, and ideal amenability be extended to Fréchet algebras?
• RQ3 What characterizations of Arens regularity hold for Fréchet algebras, particularly in the context of weighted group and semigroup algebras?
• RQ4 How do the multiplier algebras of Fréchet algebras relate to Segal algebra structures, and what topologies govern their behavior?
• RQ5 Are the Arens regularity of a Fréchet algebra and its bidual equivalent, and how does this relate to the 0-cluster property of weight functions?

Key findings

• If $\mathcal{B}$ is a semisimple Fréchet algebra and $T: \mathcal{A} \to \mathcal{B}$ is a surjective homomorphism satisfying a certain continuity condition, then $T$ is continuous, generalizing Johnson’s uniqueness of norm theorem.
• Every semisimple Banach algebra has a unique topology as a Fréchet algebra, as continuity of epimorphisms is automatic in the Banach case.
• For a commutative Fréchet $Q$-algebra $\mathcal{B}$ with approximate identity, the spaces of modular maximal closed ideals of $\mathcal{B}$ and any Segal Fréchet algebra $\mathcal{A}$ in $\mathcal{B}$ are homeomorphic.
• A Fréchet algebra $\mathcal{A}$ is Arens regular if and only if the first and second Arens products coincide on $\mathcal{A}^{**}$, and this property is preserved under bicommutative structures.
• For a locally compact group $G$ with weight $w$, $L^1(G,w)^{**}$ is Arens regular if and only if $\Omega(x,y) = \frac{w(xy)}{w(x)w(y)}$ is 0-clustered, or $L^1(G)$ is Arens regular.
• In inverse semigroups with finitely many idempotents, $\ell^1(S)$ is Arens regular if and only if $S$ is finite, and if $w$ is bounded below and $\Omega$ is 0-clustered, then $S$ is countable.