Projectivity of certain non-commutative L_p spaces as modules over Fourier algebra.

Dales and Polyakov (2004) investigated projectivity of left L1(G)-modules for locally compact group G. The class of modules include C_0(G) and L^p(G) for 1< p < ∞. They proved that C_0(G) (resp. L^p(G) for 1<p<∞) is projective iff G is compact. In this talk we focus on the dual situation, namely A(G)-modules C^*_r(G) and L^p(VN(G)) for 1<p< ∞. We will show that C^*_r(G) (resp. L^p(VN(G)) for 1<p<∞) is an operator projective left A(G)-module when G is discrete and amenable. Conversely, we can show that C^*_r(G) (resp. L^p(VN(G)) for 2≤ p < ∞) is not operator projective when G is not discrete. Unlike in the case of L1(G)-modules amenability plays an important role here. Indeed, C^*_r(G) (resp. L^p(VN(G)) for 1<p<∞) is not an projective left operator A(G)-module when G is a discrete group containing \mathbb{F}_2, the free group with two generators.