Postquantum Brègman relative entropies

We develop 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 few families of Br\`{e}gman relative entropies over several radially compact base normed spaces in spectral duality. In particular, we prove generalised pythagorean theorem and norm-to-norm continuity of the corresponding entropic projections for a family induced on preduals of any W$^*$-algebras and of semifinite JBW-algebras using Mazur maps into corresponding noncommutative and nonassociative $L_p$ spaces. We also prove generalised pythagorean theorem for a family induced using Kaczmarz maps into Orlicz spaces over semifinite W$^*$-algebras, and for a family over generalised spin factors. Additionally, we establish Lipschitz--H\"{o}lder continuity of the nonassociative Mazur map on positive parts of unit balls, characterise several geometric properties of the Morse-Transue-Nakano-Luxemburg norm on noncommutative Orlicz spaces, and introduce a new family of $L_p$ spaces over order unit spaces.