Affine connections, duality and divergences for a von Neumann algebra

On the predual of a von Neumann algebra, we define a differentiable manifold structure and affine connections by embeddings into non-commutative L_p-spaces. Using the geometry of uniformly convex Banach spaces and duality of the L_p and L_q spaces for 1/p+1/q=1, we show that we can introduce the \alpha-divergence, for \alpha in (-1,1), in a similar manner as Amari in the classical case. If restricted to the positive cone, the \alpha-divergence belongs to the class of quasi-entropies, defined by Petz.