New approach to quantum and nonassociative Brègman relative entropies

This paper presents some elements of a new approach to construction of Br\`{e}gman relative entropies over nonreflexive Banach spaces, based on nonlinear mappings into reflexive Banach spaces. We apply it to derive a new family of Br\`{e}gman relative entropies over preduals of any W$^*$-algebras and of semifinite JBW-algebras, induced using the Mazur maps into corresponding noncommutative and nonassociative $L_p$ spaces. We prove generalised pythagorean theorem and norm-to-norm continuity of the corresponding entropic projections, as well as H\"{o}lder continuity of the nonassociative Mazur map on positive parts of unit balls. We also discuss the possibility of extension of these results to base normed spaces in spectral duality, pointing to an open problem of construction of $L_p$ spaces over the corresponding order unit spaces.