Projectivity of modules over Fourier algebras

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.Comment: 32 pages, numerous typos and errors are corrected. To appear in Proc. London Math. So

PROJECTIVITY OF MODULES OVER SEGAL ALGEBRAS

Abstract. In this paper we will study the projectivity of various natural modules associated to operator Segal algebras of the Fourier algebra of a locally compact group G. In particular, we will focus on the question of identifying when such modules will be projective in the category of operator spaces. We show that projectivity often implies that the underlying group is discrete or even finite. We will also look at the projectivity for modules of Acb(G), the closure of A(G) in the space of its completely bounded multipliers. Here we give some evidence to show that weak amenability of G plays an important role. 1