  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.
