Affine connections, duality and divergences for a von Neumann algebra

2003/11/05 by Anna Jencova, Anna Jenčová, Jencova, Anna · 1 citation
Mathematics · Physics and Astronomy · #Advanced Banach Space Theory #Advanced Operator Algebra Research #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Mathematical Inequalities and Applications #Mathematical Physics (math-ph) #math-ph #math.DG #math.MP

paper · pdf · doi:10.48550/arxiv.math-ph/0311004

20 pages

arxiv created 2003/11/05 · openalex publication_date 2003/11/05 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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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