[Paper Review] Amenability properties of the central Fourier algebra of a compact group

Mahmood Alaghmandan, Nico Spronk|arXiv (Cornell University)|Jul 25, 2014

Advanced Operator Algebra Research46 references4 citations

TL;DR

This paper investigates amenability properties of the central Fourier algebra ZA(G) on a compact group G. It establishes that ZA(G) admits bounded point derivations if G has a non-abelian connected subgroup, while ZA(G) is amenable if and only if G is virtually abelian. The key contribution is a complete characterization of amenability, weak amenability, and hyper-Tauberian properties of ZA(G) in terms of the group's structure, particularly its connected component and finite quotients.

ABSTRACT

We let the central Fourier algebra, ZA(G), be the subalgebra of functions u in the Fourier algebra A(G) of a compact group, for which u(xyx^{-1})=u(y) for all x,y in G. We show that this algebra admits bounded point derivations whenever G contains a non-abelian closed connected subgroup. Conversely when G is virtually abelian, then ZA(G) is amenable. Furthermore, for virtually abelian G, we establish which closed ideals admit bounded approximate identities. We also show that if ZA(G) is weakly amenable, even hyper-Tauberian, exactly when G admits no non-abelian connected subgroup. We also study the amenability constant of ZA(G) for finite G and exhibit totally disconnected groups G for which ZA(G) is non-amenable.

Motivation & Objective

• To determine when the central Fourier algebra ZA(G) of a compact group G is amenable, weakly amenable, or hyper-Tauberian.
• To analyze the existence of bounded point derivations in ZA(G) in relation to the group's structure, especially the presence of non-abelian connected subgroups.
• To characterize closed ideals in ZA(G) that admit bounded approximate identities when G is virtually abelian.
• To compute and analyze the amenability constant AM(ZA(G)) for finite and profinite groups.
• To extend results on spectral synthesis and weak synthesis to the spectrum of ZA(G), generalizing earlier work by Meany and Ricci.

Proposed method

• Define ZA(G) as the subalgebra of A(G) invariant under inner automorphisms, i.e., functions u with u(xyx⁻¹) = u(y) for all x,y ∈ G.
• Use representation theory and harmonic analysis on compact groups to analyze the structure of ZA(G), particularly when G is virtually abelian or has non-abelian connected components.
• Apply techniques from operator space theory and Banach algebra theory, including bounded approximate identities and derivations, to study amenability and weak amenability.
• Leverage the isomorphism ZA(G×G) ≅ ZA(G) ˆ⊗ ZA(G) and the diagonal element 1_Conj(G)_D to compute the amenability constant via tensor product properties.
• Use the fact that for finite groups, AM(ZA(G)) is multiplicative over products, and apply this to infinite products of non-abelian finite groups to show non-amenability.
• Employ spectral synthesis techniques to study finite and singleton subsets of the spectrum of ZA(G), generalizing results of Meany and Ricci.

Experimental results

Research questions

• RQ1 When is the central Fourier algebra ZA(G) amenable for a compact group G?
• RQ2 Does the existence of a non-abelian connected subgroup in G imply the existence of a bounded point derivation on ZA(G)?
• RQ3 For virtually abelian compact groups, which closed ideals in ZA(G) admit bounded approximate identities?
• RQ4 When is ZA(G) weakly amenable or hyper-Tauberian?
• RQ5 What is the amenability constant AM(ZA(G)) for finite and profinite groups, and when is it infinite?

Key findings

• ZA(G) admits a bounded point derivation if G contains a non-abelian closed connected subgroup.
• ZA(G) is amenable if and only if G is virtually abelian.
• ZA(G) is weakly amenable (and hyper-Tauberian) if and only if the connected component G_e is abelian.
• For an infinite product P = ∏G_i of finite groups, ZA(P) is non-amenable if infinitely many G_i are non-abelian, as AM(ZA(P)) = ∞.
• The amenability constant satisfies AM(ZA(P)) = ∏AM(ZA(G_i)) for finite products, and AM(ZA(G)) ≥ AM(ZA(F)) when G = H × F with F finite.
• The paper establishes a broad generalization of Lasser’s amenability result and provides new spectral synthesis results for finite and singleton subsets of the spectrum of ZA(G).