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.

翻译：我们提出了一种基于非线性映射到自反Banach空间的新方法，用于在非自反Banach空间上构造Brègman相对熵。我们将该方法应用于谱对偶中多个径向紧致基赋范空间上的几类Brègman相对熵的推导。特别地，我们证明了广义毕达哥拉斯定理，并证明了由Mazur映射诱导到相应非交换和非结合$L_p$空间上的族在任意W$^*$-代数及半有限JBW-代数前对偶上的熵投影的范数-范数连续性。同时，我们证明了由Kaczmarz映射诱导到半有限W$^*$-代数上Orlicz空间中的族的广义毕达哥拉斯定理，以及广义自旋因子上的相应性质。此外，我们建立了单位球正部上非结合Mazur映射的Lipschitz-Hölder连续性，刻画了非交换Orlicz空间上Morse-Transue-Nakano-Luxemburg范数的若干几何性质，并引入了一类定义在序单位空间上的新型$L_p$空间。