[Paper Review] Weak amenability of Fourier algebras on compact groups

Brian Forrest, Ebrahim Samei+1 more|ArXiv.org|Aug 13, 2008

Advanced Operator Algebra Research18 references3 citations

TL;DR

This paper establishes that the Fourier algebra A(G) of a compact group G is weakly amenable if and only if the connected component of the identity, Gₑ, is abelian. The result is proven via connections to spectral synthesis on the anti-diagonal and hyper-Tauberian properties, and extended to non-compact groups with small invariant neighborhoods or maximal almost periodicity, resolving a long-standing conjecture in harmonic analysis on operator algebras.

ABSTRACT

We give for a compact group G, a full characterisation of when its Fourier algebra A(G) is weakly amenable: when the connected component of the identity G_e is abelian. This condition is also equivalent to the hyper-Tauberian property for A(G), and to having the anti-diagonal D^v={(s,s^{-1}):s is in G} being a set of spectral synthesis for A(GXG). We show the relationship between amenability and weak amenability of A(G), and (operator) amenability and (operator) weak amenability of A_D(G), an algebra defined by the authors in arXiv:0705.4277. We close by extending our results to some classes of non-compact, locally compact groups, including small invariant neighbourhood groups and maximally weakly almost periodic groups.

Motivation & Objective

• To resolve the conjecture that A(G) is weakly amenable if and only if the connected component Gₑ of a compact group G is abelian.
• To establish the equivalence between weak amenability of A(G), spectral synthesis of the anti-diagonal Δ̌ in A(G×G), and the hyper-Tauberian property.
• To extend the characterization of weak amenability to non-compact locally compact groups, including small invariant neighborhood groups and maximally almost periodic groups.
• To clarify the relationship between amenability, weak amenability, and operator space structures in A(G) and the related algebra AΔ(G).

Proposed method

• Proved that if A(G) is weakly amenable, then Gₑ must be abelian, using ideal-theoretic and spectral synthesis techniques.
• Established that weak amenability of A(G) is equivalent to the anti-diagonal Δ̌ being a set of local synthesis in A(G×G), leveraging regularity and ideal structure.
• Used the operator space framework to relate weak amenability of A(G) to the operator weak amenability of the ideal I′(Δ̌) in A(G×G).
• Applied results from [10] on the algebra AΔ(G), showing it captures essential structural information of A(G) for compact groups.
• Extended results to non-compact groups by analyzing open subgroups H ≅ A×K with A abelian and K compact, using invariance and automorphism invariance.
• Utilized partition of unity and local synthesis arguments to lift global properties from open subgroups to the full group G.

Experimental results

Research questions

• RQ1 When is the Fourier algebra A(G) of a compact group G weakly amenable?
• RQ2 What is the precise structural condition on G that ensures weak amenability of A(G)?
• RQ3 How are weak amenability, spectral synthesis on the anti-diagonal Δ̌, and the hyper-Tauberian property related in A(G)?
• RQ4 Can the characterization of weak amenability be extended beyond compact groups to non-compact classes like small invariant neighborhood groups?
• RQ5 What is the role of the connected component Gₑ in determining the weak amenability of A(G)?

Key findings

• For a compact group G, A(G) is weakly amenable if and only if the connected component Gₑ is abelian.
• If A(G) admits any weakly amenable ideal, then Gₑ must be abelian, showing the condition is necessary even for ideals.
• The anti-diagonal Δ̌ = {(s,s⁻¹) : s ∈ G} is a set of local synthesis for A(G×G) if and only if A(G) is weakly amenable.
• The hyper-Tauberian property of A(G) holds precisely when Gₑ is abelian, and this implies weak amenability.
• For groups G with an open subgroup H ≅ A×K (A abelian, K compact), weak amenability of A(G) is equivalent to Gₑ being abelian.
• The result extends to small invariant neighborhood groups and maximally almost periodic groups, where the same characterization holds.