Amenability and weak amenability of the Fourier algebra

Abstract

Let G be a locally compact group. We show that its Fourier algebra A(G) is amenable if and only if G has an abelian subgroup of finite index, and that its Fourier-Stieltjes algebra B(G) is amenable if and only if G has a compact, abelian subgroup of finite index. We then show that A(G) is weakly amenable if the component of the identity of G is abelian, and we prove some partial results towards the converse.

Lp-Fourier and Fourier–Stieltjes algebras for locally compact groups

“…In many ways A(G) is analagous to the group algebra L 1 (G), however it is not the case that A(G) is amenable if and only if G is amenable. In fact Brian Forrest and Volker Runde showed that A(G) is amenable if and only if G is almost abelian, i.e., if and only if G contains an open abelian subgroup of finite index [7]. Recall that A(G) is the predual of V N(G) and, hence, has a canonical operator space structure.…”

Connes-amenability of Fourier–Stieltjes algebras

“…We shall prove that B(G) is Connes-amenable if and only if G is almost abelian. The method of proof resembles in some ways the one used in [FR05] and [Run08]: from the existence of a C w σ -virtual diagonal for B(G), we conclude that a certain map is completely bounded, which is possible only if G is almost abelian. However, in comparison with [FR05] and [Run08], there are considerable technical difficulties to overcome.…”

“…It is a dual Banach algebra with natural predual C * (G), the (full) group C * -algebra of G. The natural question arises which locally compact groups G have a Connes-amenable Fourier-Stieltjes algebra B(G). With an eye on the main results of [Run03a] and [FR05], one is led to the conjecture that B(G) is Connes-amenable if and only if G is almost abelian, i.e., has an abelian subgroup of finite index (with the "if" part being fairly straightforward; see [Run04, Proposition 3.1]). There has been some circumstantial evidence suggesting that the "only if" part of the conjecture might also be true, meaning that it has been corroborated in certain special cases: it definitely holds true if G is discrete and amenable ([Run04, Theorem 3.5]) or the topological product of a family of finite groups ([Run04, Theorem 3.4]).…”

Biflatness and biprojectivity of the Fourier algebra

“…Then A(G) has a bounded approximate identity by [13], making A(G) amenable. By [6,Theorem 2.3], this means that (a) holds.…”