Amenability properties of the central Fourier algebra of a compact group (Q2362696)

Let \(G\) be a compact group. This article concerns amenability properties of a subalgebra of the Fourier algebra \(A(G)\), which the authors call the \textit{central Fourier algebra} \(ZA(G)\). Specifically, \(ZA(G)\) is the subalgebra of class functions in \(A(G)\), i.e., functions such that \(u(x) = u(yxy^{-1})\) for all \(x,y\in G\). Equivalently, \(ZA(G)\) is the predual of the centre of the von Neumann algebra of \(G\). The results presented are supposed to parallel similar results for the centre of the \(L^1\)-algebra of the group (see, e.g., [the first author and \textit{J. Crann}, ``Fourier algebras of hypergroups and central algebras on compact (quantum) groups'', Studia Math. (to appear)]. The authors show that if \(G\) is virtually abelian, then \(ZA(G)\) is amenable. Moreover, they show that \(ZA(G)\) is weakly amenable if and only if the connected component of the identity \(G_e\) is abelian. Then, necessity is obtained by showing that when \(G\) contains a closed, connected non-abelian subgroup, then \(ZA(G)\) admits bounded point derivations (see the article for the definition of a point derivation). Other related results are obtained in passing.