Biflatness and biprojectivity of the Fourier algebra

TL;DR

This paper characterizes the structural properties of the Fourier algebra of a locally compact group, showing that biflatness or biprojectivity impose strong algebraic and topological restrictions on the group.

Contribution

It establishes a dichotomy for groups based on the biflatness and biprojectivity of their Fourier algebra, linking algebraic properties to group structure.

Findings

Biflatness implies the group has an abelian subgroup of finite index or is non-amenable without F_2.

Biprojectivity leads to a similar structural dichotomy.

Provides new insights into the relationship between algebraic properties of Fourier algebras and group theory.

Abstract

We show that the biflatness - in the sense of A. Ya. Helemskii - of the Fourier algebra $A(G)$ of a locally compact group $G$ forces $G$ to either have an abelian subgroup of finite index or to be non-amenable without containing $F_2$, the free group in two generators, as a closed subgroup. An analogous dichotomy is obtained for biprojectivity.