Operator amenability of the Fourier algebra in the cb-multiplier norm

TL;DR

This paper proves that the closure of the Fourier algebra in the cb-multiplier norm is operator amenable for certain discrete groups, including free groups, revealing new properties of these algebras.

Contribution

It establishes operator amenability of $A_{cb}(G)$ for weakly amenable, residually finite-dimensional discrete groups, including non-amenable free groups.

Findings

$A_{cb}(G)$ is operator amenable for certain groups.

$A_{cb}(F_2)$ is operator amenable despite $F_2$ being non-amenable.

Characterization of weakly completely complemented ideals in $A(G)$.

Abstract

Let $G$ be a locally compact group, and let $A_\cb(G)$ denote the closure of $A(G)$, the Fourier algebra of $G$, in the space of completely bounded multipliers of $A(G)$. If $G$ is a weakly amenable, discrete group such that $\cstar(G)$ is residually finite-dimensional, we show that $A_\cb(G)$ is operator amenable. In particular, $A_\cb(F_2)$ is operator amenable even though $F_2$, the free group in two generators, is not an amenable group. Moreover, we show that, if $G$ is a discrete group such that $A_\cb(G)$ is operator amenable, a closed ideal of $A(G)$ is weakly completely complemented in $A(G)$ if and only if it has an approximate identity bounded in the cb-multiplier norm.