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