(Non-)amenability of the Fourier algebra in the cb-multiplier norm

TL;DR

This paper investigates the conditions under which the Fourier algebra's cb-multiplier norm closure is amenable, revealing deep connections between group properties and algebraic amenability.

Contribution

It establishes that if the cb-multiplier closure of the Fourier algebra is amenable, then the group's almost periodic compactification must have an abelian subgroup of finite index, linking group structure to algebraic properties.

Findings

A_cb(G) cannot be amenable if G contains a free group of two generators as a closed subgroup.

Amenability of A_{M_cb}(G) implies a specific structure of the almost periodic compactification.

The paper provides conditions that connect group properties with the amenability of associated Fourier algebras.

Abstract

For a locally compact group $G$, let $A(G)$ denote its Fourier algebra, $M_{cb}(A(G))$ the completely bounded multipliers of $A(G)$, and $A_{M_cb}(G)$ the closure of $A(G)$ in $M_{cb}(A(G))$. We show that, if $A_{M_cb}(G)$ is amenable, then $a(G_d)$, the almost periodic compactification of the discretization of $G$, has an abelian subgroup of finite index. As a consequence, $A_{M_cb}(G)$ cannot be amenable if $G$ contains a copy of $\free_2$, the free group in two generators, as a closed subgroup.