[Paper Review] Approximate weak amenability of Banach algebras

G. H. Esslamzadeh, Behrouz Shojaee|arXiv (Cornell University)|Aug 25, 2009

Advanced Operator Algebra Research9 references3 citations

TL;DR

This paper introduces and studies four generalized notions of amenability for Banach algebras: approximate, approximate weak, approximate cyclic, and approximate $n$-weak amenability. It establishes that if the second dual $$\mathcal{A}^{**}$ is approximately weakly (or $n$-weakly) amenable, then so is $$\mathcal{A}$, and proves that approximate $(n+2)$-weak amenability implies approximate $n$-weak amenability, resolving open questions on hereditary properties of approximate amenability concepts.

ABSTRACT

In this paper we deal with four generalized notions of amenability which are called approximate, approximate weak, approximate cyclic and approximate $n$-weak amenability. The first two were introduced and studied by Ghahramani and Loy in [9]. We introduce the third and fourth ones and we show by means of some examples, their distinction with their classic analogs. Our main result is that under some mild conditions on a given Banach algebra $\A$, if its second dual $\A^{**}$ is $(2n-1)$-weakly [respectively approximately/ approximately weakly/ approximately $n$-weakly] amenable, then so is $\A$. Also if $\A$ is approximately $(n+2)$-weakly amenable, then it is approximately $n$-weakly amenable. Moreover we show the relationship between approximate trace extension property and approximate weak [respectively cyclic] amenability. This answers question 9.1 of [9] for approximate weak and cyclic amenability.

Motivation & Objective

• To extend the theory of amenability in Banach algebras by introducing and analyzing new approximate notions: approximate $n$-weak and approximate cyclic amenability.
• To investigate the hereditary properties of approximate weak and cyclic amenability, addressing Question 9.1 from Ghahramani and Loy (2009).
• To establish conditions under which approximate amenability is inherited from the second dual $$\mathcal{A}^{**}$ to the algebra $$\mathcal{A}$.
• To clarify the relationship between the approximate trace extension property and approximate weak/cyclic amenability, generalizing results from Gronbaek (1993).

Proposed method

• Introduce the concepts of approximate $n$-weak amenability and approximate cyclic amenability as generalizations of classical $n$-weak and cyclic amenability.
• Use the Arens product on the second dual $$\mathcal{A}^{**}$ to analyze module actions and derive cohomological properties.
• Prove that if $$\mathcal{A}^{**}$ is approximately weakly (or $n$-weakly) amenable, then $$\mathcal{A}$ inherits this property under mild conditions.
• Construct a net of elements in the dual space that implements derivations approximately, using the approximate trace extension property.
• Apply techniques from cohomology theory, particularly the first cohomology group $H^1(\mathcal{A}, \mathcal{X})$, to characterize approximate derivations.
• Use the unitization $$\mathcal{A}^{\#}$ to relate amenability of a Banach algebra to its quotient and ideal structures.

Experimental results

Research questions

• RQ1 Under what conditions does approximate weak amenability of $$\mathcal{A}^{**}$ imply approximate weak amenability of $$\mathcal{A}$?
• RQ2 Does approximate $(n+2)$-weak amenability imply approximate $n$-weak amenability for Banach algebras?
• RQ3 What is the role of the approximate trace extension property in characterizing approximate weak and cyclic amenability?
• RQ4 How do the hereditary properties of approximate weak and cyclic amenability compare to their classical counterparts?
• RQ5 Can the inheritance of approximate amenability from $$\mathcal{A}^{**}$ to $$\mathcal{A}$ be established for odd $n$ in the context of $n$-weak amenability?

Key findings

• If $$\mathcal{A}^{**}$ is approximately weakly amenable, then $$\mathcal{A}$ is approximately weakly amenable, under mild conditions.
• Approximate $(n+2)$-weak amenability implies approximate $n$-weak amenability for any $n \geq 1$, generalizing a classical result.
• For odd $n$, if $$\mathcal{A}^{**}$ is $(2n-1)$-weakly amenable, then $$\mathcal{A}$ is approximately $n$-weakly amenable.
• The approximate trace extension property is both necessary and sufficient for passing approximate weak amenability from a quotient algebra to the original algebra.
• Approximate cyclic amenability of $$\mathcal{A}$ is characterized via the existence of a net in $$\mathcal{A}^{*}$ implementing derivations approximately.
• There exist examples of Banach algebras that are approximately cyclic amenable but not cyclic amenable, demonstrating the strictness of the approximate notion.