[Paper Review] Homomorphisms, amenability and weak amenability of Banach algebras

M‎. ‎Eshaghi Gordji|ArXiv.org|Oct 3, 2006

Advanced Operator Algebra Research7 references3 citations

TL;DR

This paper establishes equivalent conditions for amenability and weak amenability of Banach algebras using homomorphisms into other Banach algebras. It proves that a Banach algebra 𝒜 is amenable if and only if, for every injective homomorphism φ: 𝒜 → 𝒷, the first cohomology group H¹(𝒜, 𝒷*ₚ) vanishes, introducing a new characterization that links amenability to homomorphism-induced bimodules and extends to weak and super weak amenability via second duals and module extensions.

ABSTRACT

In this paper we find some necessary and sufficient conditions for a Banach algebra to be amenable or weakly amenable, by applying the homomorphisms on Banach algebras.

Motivation & Objective

• To provide new necessary and sufficient conditions for amenability of a Banach algebra using homomorphisms into other Banach algebras.
• To relate the concepts of weak amenability and super weak amenability to homomorphism-induced bimodules and cohomology vanishing.
• To generalize existing cohomological characterizations of amenability by introducing module extension Banach algebras and second dual structures.
• To investigate the implications of amenability of the second dual algebra on the amenability properties of the original algebra.

Proposed method

• Uses module extension Banach algebras X ⊕₁ 𝒜 to construct bimodules from homomorphisms φ: 𝒜 → 𝒷.
• Defines the bimodule structure on 𝒷*ₚ for a homomorphism φ: 𝒜 → 𝒷, where 𝒷*ₚ is the dual of the 𝒜-bimodule induced by φ.
• Applies cohomological techniques: H¹(𝒜, X*) = {0} characterizes amenability, and H¹(𝒜, 𝒜*) = {0} defines weak amenability.
• Employs the second dual 𝒜** with the first Arens product to extend derivations and analyze cohomology in dual spaces.
• Uses the second transpose φ** of a homomorphism φ to relate cohomology of 𝒜 to that of 𝒜**.
• Applies the notion of point derivations and their extensions to 𝒜** to study the absence of non-zero continuous point derivations.

Experimental results

Research questions

• RQ1 What conditions on a Banach algebra 𝒜 ensure that H¹(𝒜, 𝒷*ₚ) = {0} for every injective homomorphism φ: 𝒜 → 𝒷?
• RQ2 How do homomorphisms φ: 𝒜 → 𝒷 induce bimodules that characterize amenability and weak amenability?
• RQ3 What is the relationship between the super weak amenability of 𝒜** and the super weak amenability of 𝒜?
• RQ4 Under what conditions does the amenability of 𝒜** imply the amenability of 𝒜?
• RQ5 When does the vanishing of ⟨dφ(a), φ(b)⟩ + ⟨dφ(b), φ(a)⟩ for all a,b ∈ 𝒜 imply that dφ is inner?

Key findings

• A Banach algebra 𝒜 is amenable if and only if H¹(𝒜, 𝒷*ₚ) = {0} for every injective homomorphism φ: 𝒜 → 𝒷.
• If 𝒜 is such that H¹(𝒜, 𝒷*ₚ) = {0} for all injective φ: 𝒜 → 𝒷 and all derivations dφ satisfying ⟨dφ(a), φ(b)⟩ + ⟨dφ(b), φ(a)⟩ = 0 are inner, then 𝒜 is amenable.
• The second dual 𝒜** being super weakly amenable implies that 𝒜 is super weakly amenable, provided 𝒜 satisfies conditions (i)–(iii) such as being a left ideal in 𝒜** or Arens regular.
• If 𝒜** is super weakly amenable, then 𝒜 is essential and admits no non-zero continuous point derivations.
• There exists a semiweakly amenable Banach algebra 𝒜 such that 𝒜# is super weakly amenable but 𝒜 is not super weakly amenable, demonstrating a strict hierarchy in amenability properties.
• The cohomology group H¹(𝒜, (X ⊕₁ 𝒜)*) vanishes for all dual bimodules X*, which implies that every derivation D: 𝒜 → X* is inner, thus establishing amenability.