Projectivity of modules over Fourier algebras

TL;DR

This paper investigates the conditions under which modules over Fourier algebras are projective, revealing connections to the group's discreteness and amenability, and exploring their homological properties.

Contribution

It provides new insights into the projectivity of modules over Fourier algebras and links this property to the group's structural features.

Findings

Projectivity often implies the group is discrete

Amenability influences module projectivity

Homological properties relate to group structure

Abstract

In this paper we will study the homological properties of various natural modules associated to the Fourier algebra of a locally compact group. In particular, we will focus on the question of identifying when such modules will be projective in the category of operator spaces. We will show that projectivity often implies that the underlying group is discrete and give evidence to show that amenability also plays an important role.