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.

翻译：本文提出了一种基于非自反Banach空间到自反Banach空间非线性映射的Brègman相对熵构造新方法要素。我们将其应用于导出任意W$^*$-代数及半有限JBW-代数预对偶空间上新的Brègman相对熵族，该族通过Mazur映射诱导至相应的非交换与非结合$L_p$空间。我们证明了广义毕达哥拉斯定理、相应熵投影的范数-范数连续性，以及单位球正部上非结合Mazur映射的Hölder连续性。同时讨论了将这些结果推广至谱对偶中基底赋范空间的可能性，并指出了在相应序单位空间上构造$L_p$空间的开放问题。