  We prove global well-posedness for a cubic, non-local Schr\"odinger equation
with radially-symmetric initial data in the critical space $L^2(\R^2)$, using
the framework of Kenig-Merle and Killip-Tao-Visan. As a consequence, we obtain
a global well-posedness result for Schr\"odinger maps from $\R^2$ into $\S^2$
(Landau-Lifshitz equation) with radially symmetric initial data (with no size
restriction).
