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Affine connections, duality and divergences for a von Neumann algebra

Anna Jencova
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Abstract

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 α-divergence, for αin (-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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Anna Jencova. 2003-11-05. Affine connections, duality and divergences for a von Neumann algebra. https://arxiv.org/abs/math-ph/0311004

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