Projectivity of modules over Segal algebras

TL;DR

This paper investigates the conditions under which modules over operator Segal algebras of the Fourier algebra are projective, revealing connections to the group's discreteness and weak amenability.

Contribution

It characterizes the projectivity of modules over operator Segal algebras and explores the influence of group properties like weak amenability.

Findings

Projectivity often implies the group is discrete or finite.

Weak amenability of the group affects module projectivity.

Provides conditions for projectivity in modules of $A_{cb}(G)$.

Abstract

In this paper we will study the projetivity of various natural modules associated to operator Segal algebras of 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. Projectivity often implies that the underlying group is discrete or even finite. We will also look at the projectivity for modules of $A_{cb}(G)$, the closure of $A(G)$ in the space of its completely bounded mutipliers. Here we give an evidence to show that weak amenability of $G$ plays an important role.