Crossed-products extensions, of Lp-bounds for amenable actions

We will extend earlier transference results due to Neuwirth and Ricard from the context of noncommutative Lp-spaces associated with amenable groups to that of noncommutative Lp-spaces associated with crossed-products of amenable actions. Namely, if m:G→C is a completely bounded Fourier multiplier on Lp, then it extends to the crossed-product with similar bounds provided that the action θ is amenable and trace-preserving. Furthermore, our construction also allows to extend G-equivariant completely bounded operators acting on the space part to the crossed-product provided that the generalized Følner sets of the action θ satisfy certain accretivity property. As a corollary we obtain stability results for maximal Lp-bounds over crossed products. We derive, using that stability results, an application to the boundedness of smooth multipliers in the Lp-spaces of group algebras.