Projectivity of modules over Segal algebras

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.